Essentially Tight Kernels for (Weakly) Closed Graphs

Abstract

We study kernelization of classic hard graph problems when the input graphs fulfill triadic closure properties. More precisely, we consider the recently introduced parameters closure number c and weak closure number $\gamma$ (Fox et al. SIAM J Comput 49(2):448–464, 2020) in addition to the standard parameter solution size k. The weak closure number $\gamma$ of a graph is upper-bounded by the minimum of its closure number c and its degeneracy d. For Capacitated Vertex Cover, Connected Vertex Cover, and Induced Matching we obtain the first kernels of size $k^{\mathcal {O}(\gamma )}$, $k^{\mathcal {O}(\gamma )}$, and $(\gamma k)^{\mathcal {O}(\gamma )}$, respectively. This extends previous results on the kernelization of these problems on degenerate graphs. These kernels are essentially tight as these problems are unlikely to admit kernels of size $k^{o(\gamma )}$ by previous results on their kernelization complexity on degenerate graphs (Cygan et al. ACM Trans Algorithms 13(3):43:1–43:22, 2017). For Capacitated Vertex Cover, we show that even a kernel of size $k^{o(c)}$ is unlikely. In contrast, for Connected Vertex Cover, we obtain a kernel with $\mathcal {O}(ck^2)$ vertices. Moreover, we prove that searching for an induced subgraph of order at least k belonging to a hereditary graph class $\mathcal {G}$ admits a kernel of size $k^{\mathcal {O}(\gamma )}$ when $\mathcal {G}$ contains all complete and all edgeless graphs. Finally, we provide lower bounds for the kernelization of Independent Set on graphs with constant closure number c and kernels for Dominating Set on weakly closed split graphs and weakly closed bipartite graphs.

1 Introduction

A main tool for coping with hard computational problems is to shrink large input data to a computationally hard core by removing easy parts of the instance in polynomial time. Parameterized algorithmics provides the framework of kernelization for analyzing the power and limits of polynomial-time data reduction algorithms. In addition to the input instance I, a parameterized problem comes equipped with a parameter k which describes the structure of the input or is a bound on the solution size. A kernelization for a parameterized problem L is an algorithm that replaces every input instance (I, k) of L in polynomial time by an equivalent instance $\left( I',k'\right)$ of L (the kernel) whose size depends only on the parameter k, that is, $\vert I' \vert + k'\le g(k)$ for some computable function g. The kernel is guaranteed to be small if k is small and g grows only modestly. A particularly important special case is thus the class of kernelizations where g is a polynomial function. Such kernelizations are referred to as polynomial kernelizations.

Many problems are not believed to admit a polynomial kernel simply because they are believed to be not fixed-parameter tractable. That is, it is assumed that they are not solvable in $f(k)\cdot \vert I \vert ^{\mathcal {O}(1)}$ time. A classic example is Dominating Set parameterized by the solution size k. Moreover, some problems that do admit kernels are believed to not admit polynomial kernels [3, 8, 22, 23];¹ a classic example is Connected Vertex Cover parameterized by the solution size k [8].

To devise polynomial kernelizations for such problems, one considers either additional parameters or restricted classes of input graphs. One example for this approach is kernelization in degenerate graphs [5, 6, 26]. A graph G is d-degenerate if every subgraph of G contains at least one vertex with degree at most d. The degeneracy of a graph G is the smallest number d such that G is d-degenerate. Dominating Set, for example, admits a kernel of size $k^{\mathcal {O}(d^2)}$ where d is the degeneracy of the input graph [26]. In this bound, the exponent of the kernel size depends on d; to emphasize this fact we will say that Dominating Set admits a polynomial kernel on d-degenerate graphs. This kernelization was shown to be tight in the sense that there is no kernel of size $k^{o(d^2)}$ [5]. The situation is different for Independent Set which admits a trivial kernel with $\mathcal {O}(dk)$ vertices: here the kernel size is polynomial in $d+k$.

Real-world networks have small degeneracy d, making d an interesting parameter from an application point of view. Moreover, bounded degeneracy imposes a combinatorial structure that can be exploited algorithmically as evidenced by the discussion above. Recently, Fox et al. [15] discovered two new parameters that share these two features: they are well-motivated from a practical standpoint and describe interesting and useful combinatorial features of graphs. The first parameter is the closure of a graph, defined as follows.

Definition 1

([15, Definition 1.1]) Let ${{\,\mathrm{\mathsf cl}\,}}_G(v) {:}{=}\max _{w \in V(G) \setminus N[v]} \left\{ \vert N(v) \cap N(w) \vert , 0 \right\}$ denote the closure number of a vertex v in a graph G. A graph G is c-closed if ${{\,\mathrm{\mathsf cl}\,}}_G(v) < c$ for all $v \in V(G)$.

In other words, a graph is c-closed if every pair of nonadjacent vertices has at most $c-1$ common neighbors. The parameter models triadic closure in social networks, the observation that people with many common acquaintances are likely to know each other. Fox et al. [15] devised another parameter, the weak closure which relates to c-closure as degeneracy relates to maximum degree: instead of demanding a bounded closure number for every vertex, one demands that every induced subgraph has some vertex with bounded closure number.

Definition 2

([15, Definition 1.3]) A graph G is weakly $\gamma$-closed if

• there exists a weak closure ordering $\sigma {:}{=}v_1, \dots , v_n$ of the vertices of G such that ${{\,\mathrm{\mathsf cl}\,}}_{G_i}(v_i) < \gamma$ for all $i \in [n]$ where $G_i {:}{=}G[\{ v_i, \dots , v_n \}]$, or, equivalently, if

• every induced subgraph $G'$ of G has a vertex $v \in V(G')$ such that ${{\,\mathrm{\mathsf cl}\,}}_{G'}(v) < \gamma$.

The weak closure number of a graph G is the minimum integer $\gamma$ such that the graph G is weakly $\gamma$-closed.

Let G be a graph and let d, c, and $\gamma$ be the degeneracy, the closure number and the weak closure number of G. The three parameters d, c, and $\gamma$ are related as follows:

• 1. The weak closure number $\gamma$ is at most $\max (d+1,c)$.

• 2. The weak closure number $\gamma$ can be arbitrarily smaller than d as witnessed by large complete graphs.

• 3. The degeneracy d and the closure number c are incomparable as witnessed by large complete graphs (they have large degeneracy and are 1-closed) and large complete bipartite graphs where one part has size two (they are 2-degenerate and have large closure number).

• 4. The second example in 3. also shows that $\gamma$ can be much smaller than the closure number c which is very often the case in real-world data [15, 20].

Akin to degeneracy, c-closure and weak $\gamma$-closure have proven to be very useful parameters. In particular, all maximal cliques of a graph can be enumerated in $3^{\gamma /3}\cdot n^{\mathcal {O}(1)}$ time [15]. By the above discussion on the relation of $\gamma$ and d, this result thus extends the range of tractable clique enumeration instances from the class of bounded-degeneracy graphs [10] to the larger class of graphs with bounded weak closure. The clique enumeration algorithm for weakly closed graphs [15] has also been extended to the enumeration of other clique-like subgraphs [1, 20]. Recently, it was shown that Dominating Set is FPT with respect to $k+\gamma$ [25]. Concerning kernelization, in previous work, we showed that Independent Set and Induced Matching admit polynomial kernels with respect to the parameter $k+c$ and that Dominating Set admits a polynomial kernel on c-closed graphs [21]. Later, we extended the kernelization result for Independent Set to parameterization by weak closure. More generally, we showed that $\mathcal {G}$-Subgraph, where one wants to find a subgraph on at least k vertices belonging to $\mathcal {G}$, admits a kernel with $\mathcal {O}(\gamma k^2)$ vertices if $\mathcal {G}$ is closed under taking subgraphs [20]. To the best of our knowledge, this is the only known kernelization result for the weak closure parameterization.

In this work we study the kernelization of several further hard graph problems on weakly closed graphs. In a nutshell, we provide kernels for a range of problems that have not been considered on weakly closed graphs so far. Our kernels are based on several combinatorial observations on the structure of weakly closed graphs that might be of more general interest.

Our Results Building on a combinatorial lemma of Frankl and Wilson [16], we obtain a general lemma (Lemma 1) which can be used to bound the size of graphs in terms of their vertex cover number and weak closure number. Essentially, the lemma bounds for graphs G with a vertex cover S of size k and weak closure $\gamma$, the number of different neighborhoods in the independent set $I{:}{=}V(G)\setminus S$ by $k^{\mathcal {O}(\gamma )}$. Lemma 1 opens up a general strategy for obtaining kernels on weakly closed graphs: devise reduction rules that 1) bound the size of the vertex cover and 2) decrease the size of neighborhood classes. We also show that Lemma 1 can be extended to a more general notion of neighborhood types, that is, a generalization from vertices to small sets of vertices (Lemma 2).

We then show that this strategy helps in obtaining kernels on weakly closed graphs for Capacitated Vertex Cover, Connected Vertex Cover, Connected $\ell$-Component Order Connectivity (Connected $\ell$-COC), and Induced Matching all parameterized by the natural parameter solution size k. For these problems, polynomial kernels on degenerate graphs are known [5, 6, 13, 18]. Our results thus extend the class of graphs for which polynomial kernels are known for these problems. The kernels have size $k^{\mathcal {O}(\gamma )}$ (Capacitated Vertex Cover, Connected Vertex Cover, and Connected $\ell$-COC) and $(\gamma k)^{\mathcal {O}(\gamma )}$ (Induced Matching), and by previous results on degenerate graphs, the dependence on $\gamma$ in the exponent cannot be avoided [5, 6]. We complement these findings with a study of Capacitated Vertex Cover and Connected Vertex Cover on c-closed graphs. Interestingly, the kernelization complexity of the problems differs: Capacitated Vertex Cover does not admit a kernel of size $\mathcal {O}\left( k^{\frac{c-1}{2}-\epsilon }\right)$ for all $\epsilon >0$ whereas Connected Vertex Cover admits a kernel with $\mathcal {O}\left( ck^2\right)$ vertices.

Next, we study the kernelization complexity of Independent Set on c-closed graphs. We show that Independent Set does not admit a kernel of size $\mathcal {O}\left( k^{2-\epsilon }\right)$ on c-closed graphs for constant c. This complements previous kernels of size $\mathcal {O}\left( c^2k^3\right)$ [21] and $\mathcal {O}\left( \gamma ^2k^3\right)$ [20], narrowing the gap between upper and lower bound for the achievable kernel size on (weakly) closed graphs. We also obtain a lower bound of $\Omega \left( k^{4/3-\epsilon }\right)$ on the number of vertices in the graph in case of constant c and show that at least a linear dependence on c is necessary in any kernelization of Independent Set on c-closed graphs: under standard assumptions, there is no kernel of size $c^{(1-\epsilon )}\cdot k^{\mathcal {O}(1)}$. Some of our results also hold for Ramsey-type problems where one wants to find a large subgraph belonging to a class $\mathcal {G}$ containing all complete and all edgeless graphs. In this context, we observe that weakly $\gamma$-closed graphs fulfill the Erdős-Hajnal property [12] with a linear dependence on $\gamma$: there is a constant q such that every weakly $\gamma$-closed graph on $k^{q\gamma }$ vertices has either a clique of size k or an independent set of size k. We believe that this observation is of independent interest and that it will be useful in the further study of weakly $\gamma$-closed graphs.

Finally, we consider Dominating Set which admits a kernel of size $k^{\mathcal {O}(c)}$ [21] and a kernel of size $k^{\mathcal {O}(d^2)}$ [26]. Recently, it was shown that Dominating Set is also fixed-parameter tractable for the smaller parameter $k+\gamma$ [25]. However, it remains an open question whether Dominating Set admits a kernel of size $k^{f(\gamma )}$ for some function f, which would extend the class of kernelizable input graphs from degenerate and c-closed to weakly closed. We make partial progress towards answering this question by showing that there is a kernel of size $k^{\mathcal {O}(\gamma ^2)}$ on graphs with constant clique number (such as bipartite graphs) and a kernel of size $(\gamma k)^{\mathcal {O}(\gamma )}$ in split graphs. In both cases these bounds are tight in the sense that kernels of size $k^{o(d^2)}$ and of size $k^{o(c)}$ are unlikely to exist [5, 20].

Preliminaries By [n] we denote the set $\{ 1, \dots , n \}$ for some $n \in \mathbb {N}$. For a graph G, let V(G) denote its vertex set, E(G) its edge set, and $n {:}{=}\vert V(G) \vert$ the number of vertices. Let $X \subseteq V(G)$ be a vertex set. By G[X] we denote the subgraph induced by X and by $G - X {:}{=}G[V(G) \setminus X]$ we denote the graph obtained from G by removing the vertices of X. If the vertices of X are pairwise adjacent (nonadjacent), then X is a clique (an independent set, respectively). We denote by $N_G(X){:}{=}\{ y \in V(G) \setminus X \mid xy \in E(G), x \in X \}$ the open neighborhood of X and by $N_G[X]{:}{=}N_G(X) \cup X$ the closed neighborhood of X. For a vertex v, we write $N_G(x){:}{=}N_G(\{x\})$ and $N_G[x]{:}{=}N_G[\{x\}]$. The maximum degree of G is $\Delta _G {:}{=}\max _{v \in V(G)} \deg _G(v)$. A vertex with degree one is referred to as a leaf-vertex.

A matching M is a set of vertex-disjoint edges. By V(M) we denote the union of all endpoints of edges in M. A vertex set S is a vertex cover of a graph G if every edge of G has at least one endpoint in S. A vertex set S is an independent set in G if no two vertices of S are pairwise adjacent in G. A vertex set S is a clique in G if every pair of vertices of S is adjacent in G.

In the remainder of this work, we fix a weak closure ordering $\sigma$. Note that such an ordering can be computed in polynomial time [15]. We define $P_G^\sigma (v) {:}{=}\{ u \in N_G(v) \mid u \text { appears before } v \text { in } \sigma \}$ and $Q_G^\sigma (v) {:}{=}\{ u \in N_G(v) \mid u \text { appears after }v\text { in }\sigma \}$. We say that $P^\sigma _G(v)$ are prior neighbors of v and $Q^\sigma _G(v)$ are posterior neighbors of v. We omit the superscripts and subscripts when they are clear from the context. The following observation follows from the definition of weak closure.

Observation 1

For nonadjacent vertices $u, v \in V(G)$, it holds that $\vert Q(u) \cap Q(v) \vert \le \vert Q(u) \cap N(v) \vert \le \gamma - 1$.

Proof

Let $G_u$ and $G_v$ be the graph induced by the vertices that appear after u and v, respectively. We have two cases based on whether u or v appears first in the weak closure ordering $\sigma$.

Case 1 u precedes v. By definition, we have $Q_G(u) \cap Q_G(v) \subseteq Q_G(u) \cap N_G(v)$ and $Q_G(u) \cap N_G(v) = N_{G_u}(u) \cap N_{G_u}(v)$. Since $\vert N_{G_u}(u) \cap N_{G_u}(v) \vert \le {{\,\mathrm{\mathsf cl}\,}}_{G_u}(u)$, it follows from the definition of weak closure that $\vert N_{G_u}(u) \cap N_{G_u}(v) \vert \le \gamma - 1$.

Case 2 v precedes u. Clearly, $Q_G(u) \cap P_G(v) = \emptyset$. We then have $Q_G(u) \cap N_G(v) = Q_G(u) \cap (P_G(v) \cup Q_G(v)) = Q_G(u) \cap Q_G(v)$. It follows from the definition of Q that $\vert Q_G(u) \cap Q_G(v) \vert = \vert N_{G_u}(u) \cap N_{G_v}(v) \vert \le \vert N_{G_v}(u) \cap N_{G_v}(v) \vert \le {{\,\mathrm{\mathsf cl}\,}}_{G_v}(v) \le \gamma - 1$. $\square$

Finally, we state the most important definitions of parameterized complexity; for further details, refer to the standard monographs [4, 9]. A parameterized problem is fixed-parameter tractable if every instance (I, k) can be solved in $f(k) \cdot \vert I \vert ^{\mathcal {O}(1)}$ time for some computable function f. An algorithm with such a running time is an FPT algorithm. A kernelization is a polynomial-time algorithm which transforms every instance (I, k) of a parameterized problem L into an equivalent instance $\left( I', k'\right)$ of L such that $\vert I' \vert + k' \le g(k)$ for some computable function g. A compression of a parameterized problem L into a problem R is an algorithm that takes as input an instance $(x,k)\in \Sigma ^*\times \mathbb {N}$ and returns a string y in time polynomial in $\vert x \vert +k$ such that $\vert y \vert$ is bounded by some polynomial in k, and $y\in R$ if and only if $(x,k)\in L$. It is widely believed that W[t]-hard problems $(t \in \mathbb {N})$ do not admit an FPT algorithm.

2 Bounding the Size of Weakly Closed Graphs with Small Twin Sets

Frankl and Wilson [16] proved the following bound on the size of set systems where the number of different intersection sizes is bounded.

Proposition 1

([16, Theorem 11]) Let $\mathcal {F}$ be a collection of pairwise distinct subsets of [n] and let $L \subseteq \{0\}\cup [n]$ be some subset. If $\vert S \cap S' \vert \in L$ for all distinct $S, S' \in \mathcal {F}$, then $\vert \mathcal {F} \vert \in \mathcal {O}\left( n^{ \vert L \vert }\right)$.

We now use this proposition to achieve a bound on the size of weakly closed graphs when every vertex has few false twins and the size of the vertex cover is small. Herein, two vertices u and v are false twins if $N(u)=N(v)$.

Lemma 1

Let G be a weakly $\gamma$-closed graph with a vertex cover S of size k, and let $I{:}{=}V(G)\setminus S$ be the corresponding independent set. If every vertex $v \in I$ has at most $t-1$ false twins, then $\vert V(G)\vert \in t \cdot \mathcal {O}\left( 3^{\gamma /3}\cdot k^{2\gamma +3}\right)$.

Proof

We say that two vertices $v, v' \in I$ are P-equivalent, Q-equivalent, and N-equivalent if $P(v) = P\left( v'\right)$, $Q(v) = Q\left( v'\right)$, and $N(v) = N\left( v'\right)$, respectively. We show that there are $\mathcal {O}\left( 3^{\gamma /3}\cdot k^{2\gamma +3}\right)$ N-equivalence classes of vertices in I. This implies the desired bound on $\vert V(G)\vert$ since each class contains at most t vertices and since S has size k. Let $\mathcal {P}$, $\mathcal {Q}$, and $\mathcal {N}$ denote the collection of P-equivalence, Q-equivalence, and N-equivalence classes, respectively. We extend the notation of P, Q, and N to equivalence classes A by defining $P(A){:}{=}P(v)$, $Q(A){:}{=}Q(v)$, and $N(A){:}{=}N(v)$ for some $v\in A$. Since there is at most one N-equivalence class for every pair of P-equivalent and Q-equivalent classes, we have $\vert \mathcal {N} \vert \le \vert \mathcal {P} \vert \cdot \vert \mathcal {Q} \vert$. Thus, it suffices to show suitable bounds on $\vert \mathcal {P} \vert$ and $\vert \mathcal {Q} \vert$.

First, we prove that $\vert \mathcal {Q} \vert \in \mathcal {O}\left( k^{\gamma }\right)$, using the result of Frankl and Wilson (Proposition 1 [16]). Since $I\supseteq A$ is an independent set, $Q(A) \subseteq S {:}{=}V(G) \setminus I$. Moreover, for two distinct Q-equivalence classes A and $A'$, we have $\vert Q(A) \cap Q\left( A'\right) \vert < \gamma$ by Observation 1, and equivalently, $\vert Q(A) \cap Q\left( A'\right) \vert \in L$ for $L {:}{=}\{0\}\cup [\gamma - 1]$. By Proposition 1 we obtain $\vert \mathcal {Q} \vert \in \mathcal {O}\left( \vert S \vert ^{ \vert L \vert }\right) = \mathcal {O}(k^{\gamma })$.

Next, we bound the size of $\mathcal {P}$. Let $I_0 {:}{=}\{ v \in I \mid \exists u, w \in P(v) :uw \notin E(G) \}$ be the set of vertices in I with nonadjacent prior neighbors. By the definition of weak $\gamma$-closure, there are at most $\gamma - 1$ vertices of $I_0$ for every pair of nonadjacent vertices in S. Thus, we have $\vert I_0 \vert < \gamma \left( {\begin{array}{c} \vert S \vert \\ 2\end{array}}\right) \in \mathcal {O}\left( \gamma k^2\right)$.

Let $I_1 {:}{=}I \setminus I_0$ and let $\mathcal {P}_1$ be the collection of P-equivalence classes in $I_1$. Note that for every $A \in \mathcal {P}_1$, its neighborhood P(A) is a clique. Since a weakly $\gamma$-closed graph on n vertices has $\mathcal {O}\left( 3^{\gamma / 3} n^2\right)$ maximal cliques [15], there are $\mathcal {O}\left( 3^{\gamma / 3} k^2\right)$ equivalence classes A such that P(A) constitutes a maximal clique in G[S]. Consider an equivalence class A such that $P(A) \subset C$ for some maximal clique C in G[S]. We will show that there are $\mathcal {O}\left( 3^{\gamma /3}k^{\gamma +3}\right)$ such equivalence classes. Let u be the first vertex of $C \setminus P(A)$ in the weak closure ordering $\sigma$. Since $P(A) \subset C \subseteq N(u) = P(u) \cup Q(u)$, we have $P(A) = (P(A) \cap P(u)) \cup (P(A) \cap Q(u))$. As $P(A) \cap P(u) = C \cap P(u)$ by the choice of u, we can rewrite $P(A) = (C \cap P(u)) \cup B$, where $B {:}{=}P(A) \cap Q(u)$. Thus, there is at most one equivalence class of $\mathcal {P}_1$ for every maximal clique C in G[S], vertex $u \in S$, and vertex subset $B \subseteq S$, and thereby, we have $\vert \mathcal {P}_1 \vert \in \mathcal {O}\left( 3^{\gamma / 3} k^2 \cdot k \cdot b\right)$, where b denotes the number of choices for B. Observe that $P(A) = P(v)$ for some vertex $v \in I_1$ and thus that $B = Q(u) \cap P(v) \subseteq Q(u) \cap N(v)$. Recall that u and v are not adjacent by the choice of u. It follows that $\vert B \vert \le \vert Q(u) \cap N(v) \vert \le \gamma - 1$ by Observation 1, and hence $b \in \mathcal {O}\left( k^{\gamma }\right)$ and $\vert \mathcal {P}_1 \vert \in \mathcal {O}\left( 3^{\gamma /3}\cdot k^3 \cdot k^{\gamma }\right) = \mathcal {O}\left( 3^{\gamma /3}\cdot k^{\gamma +3}\right)$. Overall, we have $\vert \mathcal {P} \vert \le \left( \vert I_0 \vert + \vert \mathcal {P}_1 \vert \right) \in \mathcal {O}\left( 3^{\gamma /3}\cdot k^{\gamma +3}\right)$. The total number of N-equivalence classes is thus at most $\vert \mathcal {Q} \vert \cdot \vert \mathcal {P} \vert \in \mathcal {O}\left( 3^{\gamma /3}\cdot k^{2\gamma +3}\right)$. $\square$

We now show that Proposition 1 can be also extended to bound the number of vertices in terms of the $\ell$-COC number. This is the smallest size of a vertex set S such that every connected component in $G-S$ has size at most $\ell$, where $\ell$ is a fixed constant; the 1-COC number is the vertex cover number. To obtain the more general bound, we extend the notion of twins.

Definition 3

Let $G=(V,E)$ be a graph and let $A,B\subseteq V(G)$ such that $\vert A \vert = \vert B \vert =\ell$. The sets A and B are $\ell$-twins if there exists an ordering $a_1, \ldots , a_\ell$ of A and an ordering $b_1, \ldots , b_\ell$ of B such that $N(a_i)\setminus A=N(b_i)\setminus B$ for each $i\in [\ell ]$.

Note that u and v are false twins if and only if $\{ u \}$ and $\{ v \}$ are 1-twins.

Lemma 2

Let G be a weakly $\gamma$-closed graph and let S be a size-k set such that each connected component in $G-S$ has size at most $\ell$, where $\ell$ is constant. Suppose that for every connected component A in $G-S$, there are at most $t-1$ other connected components B in $G-S$ such that A and B are $\vert A \vert$-twins. Then, $\vert V(G) \vert \in \mathcal {O}( t \cdot k^{\mathcal {O}(\gamma )})$.

Proof

Let $D {:}{=}V(G) \setminus S$. In the following, let Z be a fixed graph with $z\le \ell$ vertices. Furthermore, let A and B be two connected components in D which are isomorphic to Z with the isomorphism f which maps the vertices $a_1,\ldots , a_{ z }$ of A to the vertices $b_1,\ldots , b_{z}$ of B. In the following, we say that A and B are $P_Z^f$-equivalent, $Q_Z^f$-equivalent, and $N_Z^f$-equivalent if $P(a_i)\setminus A = P(f(a_i))\setminus B$, $Q(a_i)\setminus A = Q(f(a_i))\setminus B$, and $N(a_i)\setminus A = N(f(a_i))\setminus B$, respectively. Note that since A and B are connected components in G[D], all the above-defined sets are contained in S. Let $\mathcal {P}_Z^f$, $\mathcal {Q}_Z^f$, and $\mathcal {N}_Z^f$ denote the collection of all $P_Z^f$-equivalence, $Q_Z^f$-equivalence, and $N_Z^f$-equivalence classes, respectively. Furthermore, by $\mathcal {N}$ we denote the union of all classes $\mathcal {N}_Z^f$ for some graph Z with at most $\ell$ vertices and some isomorphism f between two graphs which are isomorphic to Z. Since there is at most one $N_Z^f$-equivalence class for every $P_Z^f$-equivalence class and $Q_Z^f$-equivalence class, we have $\left| \mathcal {N}_Z^f \right| \le \left| \mathcal {P}_Z^f \right| \cdot \left| \mathcal {Q}_Z^f \right|$. Furthermore, since there exist at most $2^{\ell ^2}$ graphs with at most $\ell$ vertices and since there are at most $\ell !$ isomorphism for each fixed graph with at most $\ell$ vertices, we conclude that $\mathcal {N}\le \ell !\cdot 2^{\ell ^2} \left| \mathcal {N}_Z^f \right|$ for a graph Z with at most $\ell$ vertices and an isomorphism f which maximizes $\left| \mathcal {N}_Z \right|$. By the assumption that there are at most t vertices in each $N_Z^f$-equivalence class, we also have $\vert D \vert \le t \cdot \left| \mathcal {N} \right|$. Since $\ell$ is a constant, it thus suffices to show that $\left| \mathcal {P}_Z^f \right| , \left| \mathcal {Q}_Z^f \right| \in k^{\mathcal {O}(\gamma )}$ for each graph Z with at most $\ell$ vertices and each corresponding isomorphism f. Now, the proof works analogously to the proof of Lemma 1.

First, we prove that $\left| \mathcal {Q}_Z^f \right| \in k^{\mathcal {O}(\gamma )}$, using the result of Frankl and Wilson (Proposition 1). To this end, we consider two distinct equivalence classes X and Y in $\mathcal {Q}_Z^f$, let A be a connected component of G[D] which is associated with the equivalence class X, and let B be a connected component of G[D] which is associated with the equivalence class Y. Furthermore, let $a_1, \ldots , a_{z}$ be a fixed ordering of the vertices of A. Since X and Y are distinct, we conclude by the definition of $Q_Z^f$-equivalence that there exists an index $i\in [ z ]$ such that for $A_i{:}{=}Q(a_i)\setminus A$ and $B_i{:}{=}Q\left( f(a_i)\right) \setminus B$ we have $A_i\ne B_i$. By the definition of weak-closure we observe that $\vert A_i\cap B_i \vert <\gamma$. Hence, $\vert A_i\cap B_i \vert \in L$ for $L\in [\gamma -1]$. Consequently, by Proposition 1, we have $\mathcal {O}\left( \vert S \vert ^{\vert L \vert }\right) = \mathcal {O}\left( k^{\gamma }\right)$ possibilities for different neighborhoods in S of the i-th vertex in A. Since $\vert A \vert = z \le \ell$ and $\ell$ is a constant, there exist $\mathcal {O}\left( k^{\gamma \cdot \ell }\right) =k^{\mathcal {O}(\gamma )}$ equivalence classes in $\mathcal {Q}_Z^f$.

Second, we show that $\vert \mathcal {P}_Z^f \vert \in k^{\mathcal {O}(\gamma )}$. Let $D_0 {:}{=}\{ v \in D \mid \exists u, w \in (P(v)\cap S) :uw \notin E(G) \}$ be the set of vertices in D with nonadjacent prior neighbors in S. By the definition of weak $\gamma$-closure, there are at most $\gamma - 1$ vertices of $D_0$ for every pair of nonadjacent vertices in S. Thus, we have $\vert D_0 \vert < \gamma \left( {\begin{array}{c} \vert S \vert \\ 2\end{array}}\right) \in \mathcal {O}\left( \gamma k^2\right)$. Hence, also the number of $\mathcal {P}_Z^f$-equivalence classes in which there exists a vertex with nonadjacent prior neighbors is bounded by $\mathcal {O}\left( \gamma k^2\right)$.

Let $D_1{:}{=}D\setminus D_0$ be the set of vertices of D where all prior neighbors form a clique. Next, we bound the number of equivalence classes in $\mathcal {P}_Z^f$. Since a weakly $\gamma$-closed graph on n vertices has $\mathcal {O}\left( 3^{\gamma / 3} n^2\right)$ maximal cliques [15], there are $\mathcal {O}\left( 3^{\gamma / 3} k^2\right)$ equivalence classes $P_Z^f$ such that each graph A in that class contains a vertex v such that $P(v)\setminus A$ constitutes a maximal clique in G[S]. Now, it remains to bound the number of equivalence classes in $\mathcal {P}_Z^f$ such that for each graph A in that class and each vertex $a_i\in A$ for $i\in [z]$, the set $A_i{:}{=}P(a_i)\setminus A$ is a proper subset of some maximal clique $C_i\subseteq S$. Formally, $A_i\subset C_i\subseteq S$. Let $u_i\in C_i$ be the first vertex in the weak closure ordering $\sigma$. Since $A_i\subseteq C_i\subseteq N(u_i)=P(u_i)\cup Q(u_i)$, we have $A_i=(A_i\cap P(u_i))\cup (A_i\cap Q(u_i))$. Since $A_i\cap P(u_i)=C_i\cap P(u_i)$ by the definition of $u_i$, we can rewrite $A_i=(C_i\cap P(u_i))\cap B_i$, where $B_i{:}{=}A_i\cap Q(u_i)$. Hence, there is at most one different neighborhood in S for every maximal clique $C_i$ in G[S], vertex $u_i\in S$, and vertex set $B_i\subseteq S$ and thereby, we have $\mathcal {O}\left( 3^{\gamma /3}\right) k^2\cdot k\cdot b_i$ many of these neighborhoods, where $b_i$ denotes the number of choices for $B_i$. Since $A_i=P(a_i)\setminus A\subseteq N(a_i)$, we conclude that $B\subseteq Q(u_i)\cap N(a_i)$. By the definition of weak $\gamma$-closure we observe that $\vert B \vert \le \vert Q(u_i)\cap N(a_i) \vert <\gamma$. Hence, $b_i\in k^{\mathcal {O}(\gamma )}$. Overall, for each vertex $a_i\in A$ we have $k^{\mathcal {O}(\gamma )}$ different neighborhoods. Thus, the number of such equivalence classes $\mathcal {P}_Z^f$ is $k^{\mathcal {O}(\gamma )\cdot \ell }=k^{\mathcal {O}(\gamma )}$ since $\ell$ is a constant. $\square$

3 Applications of our Framework

We now apply Lemmas 1 and 2 to obtain kernels for several well-known problems.

3.1 Capacitated Vertex Cover

The first problem to which we apply Lemma 1 is Capacitated Vertex Cover.

Capacitated Vertex Cover admits a kernel with $\mathcal {O}\left( k^{d + 1}\right)$ vertices. This kernel is essentially tight: a kernel with $\mathcal {O}\left( k^{d-\epsilon }\right)$ vertices would imply coNP $\subseteq$ NP/poly [5]. We will show that the reduction rule used to obtain a kernel on degenerate graphs also leads to a kernel on graphs with bounded weak closure. One may view this result as a way of showing that the rules are more powerful than what was previously known. The kernel uses the following rule.

Reduction Rule 1

([5, Rule 3]) If $S \subseteq V(G)$ is a subset of false twin vertices with a common neighborhood N(S) such that $\vert S \vert = k + 2 \ge \vert N(S) \vert$, then remove a vertex with minimum capacity in S from G, and decrease all the capacities of vertices in N(S) by one.

We omit the proof for the correctness of Reduction Rule 1, referring to Cygan et al. [5, Lemma 20]. Removing a vertex v from a graph does not increase the weak $\gamma$-closure since $\sigma$ without v is a weak $\gamma$-closure ordering for $G-\{v\}$ where $\sigma$ is a weak $\gamma$-closure ordering for G.

In the following theorem, we show that Reduction Rule 1 indeed gives us a kernel with $k^{\mathcal {O}(\gamma )}$ vertices.

Theorem 1

Capacitated Vertex Cover has a kernel with $\mathcal {O}\left( 3^{\gamma /3}\cdot k^{2\gamma +4}\right)$ vertices.

Proof

We show that a Yes-instance which is reduced with respect to Reduction Rule 1 has $\mathcal {O}\left( 3^{\gamma /3}\cdot k^{2\gamma +4}\right)$ vertices. Let S be a capacitated vertex cover of size at most k of $(G,{{\,\mathrm{\mathsf cap}\,}},k)$. Let $I{:}{=}V(G)\setminus S$. By definition, I is an independent set and $N(v)\subseteq S$ for all $v\in I$. Moreover, since $(G,{{\,\mathrm{\mathsf cap}\,}},k)$ is reduced with respect to Reduction Rule 1 there is no set of $k+2$ vertices in I that have the same neighborhood. Hence, G, S and I fulfill the condition of Lemma 1 with $t=k+2$. Thus, $\vert V(G) \vert \in k\cdot \mathcal {O}\left( 3^{\gamma /3}\cdot k^{2\gamma +3}\right) = \mathcal {O}\left( 3^{\gamma /3}\cdot k^{2\gamma +4}\right)$. $\square$

We also show that this kernel is essentially tight even if $\gamma$ is replaced by c.

Theorem 2

For $c \ge 4$, Capacitated Vertex Cover  has no kernel of size $\mathcal {O}\left( k^{\frac{c-1}{2}-\epsilon }\right)$ unless coNP $\subseteq$ NP/poly.

Proof

We will show the theorem by a reduction from $\lambda$-Exact Set Cover.

We call such a set $\mathcal {S}$ an exact cover of U. For any $\lambda \ge 3$, $\lambda$-Exact Set Cover does not have a compression of size $\mathcal {O}(k^{\lambda -\epsilon })$ unless coNP $\subseteq$ NP/poly [7, 17]. Let $(U,\mathcal {F},k)$ be an instance of $\lambda$-Exact Set Cover. We will construct a $(2\lambda +1)$-closed graph G as follows: The vertex set V(G) consists of one copy of $\mathcal {F}$, and two copies $U^1$ and $U^2$ of U. Furthermore, one leaf-vertex is attached to each vertex in $U^1\cup U^2$. The edges between $\mathcal {F}$ and both copies $U^1$ and $U^2$ of U represent the incidence graph of the instance $(U,\mathcal {F},k)$. Furthermore, we add edges such that $U^1\cup U^2$ forms a clique in G. Note that G contains exactly $2\lambda k +2\lambda \vert \mathcal {F} \vert + \lambda k (2\lambda k-1)$ edges. Next, we set the capacity of each leaf-vertex which is attached to a vertex in $U^1\cup U^2$ to zero, and the capacity of each vertex in $\mathcal {F}$ to $2\lambda$. For an element $u_i\in U$, we denote by $z_i$ the number of sets in $\mathcal {F}$ containing element $u_i$. We set the capacity of the i-th vertex $u^1_i$ of $U^1$ to $z_i+2\lambda k-i$ and the capacity of the i-th vertex $u^2_i$ of $U^2$ to $z_i+i-1$. Finally, we set $k'{:}{=}2\lambda k+k$. Since $\deg (F)=2\lambda$ for each $F\in \mathcal {F}$, each leaf-vertex has degree one, and $U^1\cup U^2$ is a clique, we observe that G is indeed $2\lambda +1$-closed.

To avoid heavy notation we will not give an exact definition of the mapping function f. Instead, we say that an edge uv is mapped to one of its endpoints u. Formally, this means that $f(uv)=u$. We now show that $\mathcal {F}$ contains an exact cover of U of size k if and only if G has a capacitated vertex cover of size $2\lambda k+k$.

Let $\mathcal {S}\subseteq \mathcal {F}$ be an exact cover of U. We prove that $X{:}{=}\mathcal {S}\cup U^1\cup U^2$ is a capacitated vertex cover of G. Since $N\left[ U^1\cup U^2\right] =V(G)$, the set X is a vertex cover of size exactly $2\lambda k+k$ of G. It remains to show that there is a mapping from each edge to one of its endpoints in X. Edges between vertices in $\mathcal {F}$ and $U^1$ are mapped to the vertex in $\mathcal {F}$ if and only if this vertex is contained in $\mathcal {S}$. Otherwise, this edge is mapped to the corresponding vertex in $U^1$. We map edges between $\mathcal {F}$ and $U^2$ analogously. Edges between $U^1$ and $U^2$ are mapped to the vertex in $U^1$ and edges where one endpoint is a leaf-vertex are mapped to the other endpoint of that edge. Furthermore, an edge between two vertices in $U^1$ is mapped to the vertex with lower index and, similarly, an edge between two vertices in $U^2$ is mapped to the vertex with higher index. Hence, each vertex in $\mathcal {S}$ covers exactly $2\lambda$ edges and since $\mathcal {S}$ is an exact cover, exactly one edge incident with a vertex in $U^1\cup U^2$ is covered by $\mathcal {S}$. Thus, the i-th vertex $u^1_i\in U^1$ covers exactly $\lambda k-i$ edges to other vertices in $U^1$, exactly one edge to a leaf-vertex, exactly $\lambda k$ edges to vertices in $U^2$, and exactly $z_i-1$ edges to vertices in $\mathcal {F}$. These are exactly $z_i+2\lambda k-i$ many. Similarly, the i-th vertex $u^2_i\in U^2$ covers exactly $i-1$ edges to other vertices in $U^2$, exactly one edge to a leaf-vertex, and exactly $z_i-1$ edges to vertices in $\mathcal {F}$. These are exactly $z_i+i-1$ many. Hence, $\mathcal {S}\cup U^1\cup U^2$ is indeed a capacitated vertex cover of G.

Conversely, suppose that G has a capacitated vertex cover X of size $2\lambda k+k$. Since the capacity of each leaf-vertex is zero, we observe that $U^1\subseteq X$, and $U^2\subseteq X$. Hence, for the set $\mathcal {S}{:}{=}\mathcal {F}\cap X$ we can assume that $\vert \mathcal {S} \vert =k$. We show that $\mathcal {S}$ is an exact cover of U.

Recall that for an element $u_i$ the number $z_i$ denotes the number of sets in $\mathcal {F}$ containing $u_i$, and that each set $F\in \mathcal {F}$ has size $\lambda$. Hence, $\sum _{i=1}^{\lambda k}z_i=\lambda \vert \mathcal {F} \vert$. Furthermore, since each vertex in $\mathcal {F}$ has capacity exactly $2\lambda$, each vertex $u^1_i\in U_1$ has capacity $z_i+2\lambda k-i$, and each vertex $u^2_i\in U_2$ has capacity $z_i+i-1$, we observe that the total capacity of all vertices in X is

$$\begin{aligned} k\cdot 2\lambda +\sum _{i=1}^{\lambda k}\left( z_i+2\lambda k-i\right) +\sum _{i=1}{\lambda k}\left( z_i+i-1\right) = 2\lambda \vert \mathcal {F} \vert +2\lambda k+ \lambda k \left( 2\lambda k-1\right) . \end{aligned}$$

Since this matches the number of edges in G, we conclude the following.

Observation 2

The number of edges mapped to a vertex in X is equal to its capacity. In particular, each edge with one endpoint in $S\in \mathcal {S}$ is mapped to S.

We say that an element $u_i$ is covered $\ell$ times if $u_i\in S$ for exactly $\ell$ distinct sets $S\in \mathcal {S}$. In this terminology, we want to show that each element of U is covered exactly once by $\mathcal {S}$.

In the following, we prove inductively over the vertex indices $i\in [\lambda k]$ that

• (a) Each element $u_j$, $j\in [i]$, is covered exactly once, and that

• (b) For each index $j\in [i]$, any edge $u^1_ju^1_\ell$ in $U^1$ is mapped to the endpoint with lower index and any edge $u^2_ju^2_\ell$ in $U^2$ is mapped to the endpoint with higher index.

Afterwards, by setting $i{:}{=}\lambda k$ we can conclude that $\mathcal {S}$ is an exact cover for U. Assume towards a contradiction of a) that element $u_i$ is not covered exactly once by $\mathcal {S}$.

Case 1 Element $u_i$ is not covered by $\mathcal {S}$. We give a lower bound on the number of edges in G which have to be mapped to vertex $u^2_i$. Since $u_i$ is not covered by $\mathcal {S}$, all $z_i$ edges with the endpoint $u^2_i$ and the other endpoint in $\mathcal {F}$ have to be mapped to $u^2_i$. Furthermore, since each leaf-vertex has capacity zero, also the edge from the leaf-vertex attached to $u^2_i$ has to be mapped to $u^2_i$. First, assume that $i=1$. Then, at least $z_1+1$ edges are mapped to vertex $u^2_1$, contradicting the fact that vertex $u^2_1$ has capacity $z_1$. Second, assume that $i\ge 2$. According to the induction hypothesis b), for each $j<i$ the edge $u^2_iu^2_j$ in $U_2$ is mapped to $u^2_i$. Hence, overall at least $z_i+i$ edges are mapped to $u^2_i$. This is not possible since vertex $u^2_i$ has capacity $z_i+i-1$.

Case 2 Element $u_i$ is covered at least twice by $\mathcal {S}$. We upper-bound the number of edges in G which can be mapped to vertex $u^1_i$. Since each leaf-vertex has capacity zero, the edge from the leaf-vertex attached to $u^1_i$ is mapped to vertex $u^1_i$. Furthermore, all edges between $u^1_i$ and vertices of $U^2$ can be mapped to $u^1_i$. Also, since $u_i$ is covered at least twice by $\mathcal {S}$, according to Observation 2, we observe that at most $z_i-2$ edges between $\mathcal {F}$ and vertex $u^1_i$ can be mapped to $u^1_i$. First, assume that $i=1$. Observe that all $\lambda k-1$ many edges with one endpoint $u^1_1$ and the other endpoint in $U^1$ can be mapped to $u^1_1$. Hence, at most $1+\lambda k+z_1-2+\lambda k-1=z_1+2\lambda k -2$ edges can be mapped to $u^1_1$. According to Observation 2 this is not possible, since vertex $u^1_1$ has capacity $z_1+2\lambda k-1$. Second, assume that $i\ge 2$. According to the induction hypothesis b) for each $j<i$ the edge $u^1_iu^1_j$ in $U_1$ is mapped to $u^1_j$. Hence, at most $\lambda k-i$ edges within $U^1$ are mapped to vertex $u^1_i$. Thus, overall at most $1+\lambda k+z_i-2+\lambda k-i=z_i+2\lambda k-i-1$ many edges are mapped to $u^1_i$. Since the capacity of $u^1_i$ is exactly $z_i+2\lambda k -i$ this contradicts Observation 2.

Hence, element $u_i$ is covered exactly once. Next, we show that this implies b). To match the capacity $z_i+2\lambda k -i$ of vertex $u^1_i$, the edge from the leaf-vertex attached to $u^1_i$, all edges between $u^1_i$ and a vertex of $U_2$, and because of a) exactly $z_i-1$ edges between vertex $u^1_i$ and a vertex of $\mathcal {F}$ have to be mapped to vertex $u^1_i$. These are exactly $z_i+\lambda k$ many. By induction hypothesis of b) we know that all edges of the form $u^1_iu^1_j$ with $j<i$ are mapped to vertex $u^1_j$. Hence, exactly $\lambda k -i$ edges within $U^1$ are not mapped yet. These edges have the form $u^1_iu^1_j$ for $j\in [i+1,\lambda k]$. Because of Observation 2 all these edges have to be mapped to vertex $u^1_i$. By similar arguments it follows that all edges of the form $u^2_iu^2_j$ for $j\in [i+1,\lambda k]$ are mapped to $u^2_j$. Hence, b) is proved. As mentioned above, by setting $i=\lambda k$ we conclude from a) that $\mathcal {S}$ is an exact cover for $(U,\mathcal {F},k)$.

Observe that since $\lambda$ is a constant we obtain $k'\in \mathcal {O}(k)$. Thus, it follows from the result of Hermelin and Wu [17] that if Capacitated Vertex Cover admits a kernel of size $\mathcal {O}\left( k^{(c-1)/2-\epsilon }\right)$ for some $\epsilon >0$ in c-closed graphs, then c-Exact Set Cover admits a kernel of size $\mathcal {O}\left( k^{c -\epsilon }\right)$, implying that coNP $\subseteq$ NP/poly [7, 17]. $\square$

3.2 Connected Vertex Cover

We now provide kernels for Connected Vertex Cover , a well-studied variant of Vertex Cover which is notorious for not admitting a polynomial kernel when parameterized k [8].

We apply Lemma 1 to obtain a kernel of size $k^{\mathcal {O}(\gamma )}$. This kernel is complemented by a polynomial kernel for the parameter $k+c$. Thus, Connected Vertex Cover is very different from Capacitated Vertex Cover concerning the kernelization complexity on c-closed graphs.

A Polynomial Kernel on Weakly Closed Graphs We now use the following known rule.

Reduction Rule 2

([5, Rule 2]) If $S \subseteq V(G)$ is a set of at least two twin vertices with a common neighborhood N(S) such that $\vert S \vert > \vert N(S) \vert$, then remove one vertex v of S from G.

After exhaustive application of Reduction Rule 2 we have, again by Lemma 1 and setting $t=k+1$, that every Yes-instance has size $k^{\mathcal {O}(\gamma )}$. The proof is completely analogous to that of Theorem 1.

Theorem 3

Connected Vertex Cover admits a kernel of size $\mathcal {O}\left( 3^{\gamma /3}\cdot k^{2\gamma +4}\right)$.

This kernel is essentially tight, because there is no kernel of size $k^{o(d)}$ [5].

A Polynomial Kernel for $k+c$ In order to obtain a polynomial kernel for Connected Vertex Cover parameterized by $k+c$, let us introduce an ”annotated” version of Connected Vertex Cover defined as follows. In a red and white graph G, the vertex set is partitioned into two sets: the red and white vertices, which we denote by $V_R(G)$ and by $V_W(G)$, respectively. The annotated version of Connected Vertex Cover imposes the additional constraint that all the red vertices must be included into the solution.

We call such a set S a solution of the instance.

The first reduction rule takes care of several trivial cases; the correctness is obvious.

Reduction Rule 3

• 1. Remove all isolated white vertices.

• 2. If G has two connected components that contain edges, then return No.

• 3. If G has two connected components with red vertices, then return No.

• 4. If G has a solution of size at most one, then return Yes.

We say that a vertex $v \in V(G)$ is simplicial if its neighborhood forms a clique. In particular, any leaf-vertex is simplicial. We remove simplicial vertices in the next rule.

Reduction Rule 4

If there is a simplicial vertex $v \in V(G)$, then do the following:

• If $v \in V_R(G)$, then decrease k by 1.

• If $v \in V_W(G)$ or $\deg _G(v) = 1$, then color all the vertices in $N_G(v)$ red.

• Remove v.

Lemma 3

Reduction Rule 4 is correct.

Proof

Let v be a simplicial vertex in G and let $\left( G', k'\right)$ be the instance obtained by applying Reduction Rule 4. Suppose that (G, k) is a Yes-instance with a solution S. If $v \notin S$, then v is white by definition and hence $k' = k$. Consequently, S is a solution of $\left( G', k'\right)$. So assume that $v \in S$.

We first show that if v is red, then $S' = S \setminus \{ v \}$ is a solution of $\left( G', k'\right)$. This is clear for the case $\deg _G(v) > 1$. To see why, note that v is simplicial in G[S] and thus S remains connected after deleting v. Suppose that $\deg _G(v) = 1$. Let u be the neighbor of v. Since Reduction Rule 4 colors u red, we have to show that $u \in S$. Observe that $S \setminus \{ u,v \}$ contains at least one vertex by Step 4 of Reduction Rule 3. Thus, S must include u in order for G[S] to be connected.

Now, consider the case that v is white. Suppose that there exists a vertex $u \in N_G(v)$ with $u \notin S$. Since $N_G(v)$ is complete in G, all the vertices of $N_G(v) \setminus \{ u \}$ are included in S. It follows that $(S \setminus \{ v \}) \cup \{ u \}$ is also a solution of (G, k). Hence, we can assume that $S \supseteq N_G(v)$, thereby showing that $S \setminus \{ v \}$ is a solution of $\left( G', k'\right)$.

Conversely, suppose that $S'$ is a solution of $\left( G', k'\right)$. If v is white, $S'$ is also a solution of (G, k), because all the neighbors of v are colored red by Reduction Rule 4. Otherwise, $S' \cup \{ v \}$ is a vertex cover of size $k' + 1 \le k$. Moreover, $S' \cup \{ v \}$ is connected in G because $S'$ includes a neighbor of v: If $\deg _G(v) = 1$, then the sole neighbor u of v is red in $G'$ due to Reduction Rule 4. Otherwise, $S'$ contains at least $\vert N_G(v) \vert - 1 \ge 1$ vertices of $N_G(v)$, because $N_G(v)$ forms a clique in $G'$. $\square$

Now, we show that Reduction Rules 3 and 4 yield a polynomial kernel for Annotated Connected Vertex Cover on c-closed graphs.

Lemma 4

Annotated Connected Vertex Cover has a kernel with $\mathcal {O}\left( ck^2\right)$ vertices.

Proof

We claim that an Annotated Connected Vertex Cover instance is a No-instance if it has at least $k + c \left( {\begin{array}{c}k\\ 2\end{array}}\right)$ vertices after Reduction Rules 3 and 4 are exhaustively applied. Suppose that an instance (G, k) of Annotated Connected Vertex Cover is a Yes-instance with a solution S. Observe that because of Steps 1 to 3 of Reduction Rule 3, the graph G contains exactly one connected component. It holds for each vertex $v \in V(G) \setminus S$ that $N_G(v) \subseteq S$ since S is a vertex cover and that $\deg _G(v) \ge 1$ by Step 1 of Reduction Rule 3. Moreover, each vertex $v \in V(G) \setminus S$ must have at least two nonadjacent neighbors in S by Reduction Rule 4. It follows that $\vert V(G) \setminus S \vert < c \left( {\begin{array}{c} \vert S \vert \\ 2\end{array}}\right) \le c \left( {\begin{array}{c}k\\ 2\end{array}}\right)$ and thus $\vert V(G) \vert = \vert S \vert + \vert V(G) \setminus S \vert < k + c \left( {\begin{array}{c}k\\ 2\end{array}}\right)$. $\square$

Finally, we can reduce from Annotated Connected Vertex Cover to Connected Vertex Cover by attaching a leaf-vertex to each red vertex.

Theorem 4

Connected Vertex Cover has a kernel with $\mathcal {O}\left( ck^2\right)$ vertices.

Proof

Let (G, k) be an instance of Connected Vertex Cover . We construct an Annotated Connected Vertex Cover instance (H, k) where all the vertices in G are colored white (that is, $V_R(H) = \emptyset$, $V_W(H) = V(G)$, and $E(H) = E(G)$). By Lemma 4, we obtain an instance of $\left( H', k'\right)$ with $\mathcal {O}\left( ck^2\right)$ vertices in polynomial time. Let $G'$ be the graph obtained from $H'$ by attaching a leaf-vertex $\ell _v$ to each red vertex v.

If there is a connected vertex cover in $H'$ including all the red vertices, then it is also a connected vertex cover in $G'$. Conversely, suppose that S is a connected vertex cover of $G'$.

Then, we can assume that S contains all red vertices of $H'$. So S is a solution of $\left( H', k'\right)$. $\square$

This result stands in contrast to Capacitated Vertex Cover , which has no kernel of size $k^{o(c)}$ unless coNP $\subseteq$ NP/poly(Theorem 2).

3.3 An Extension to Connected $\ell$-COC

In Connected $\ell$-COC, the task is to find a set S of at most k vertices such that G[S] is connected and every connected component of $G - S$ has size at most $\ell$, where $\ell$ is a fixed constant. We show that this problem also admits a kernel of size $k^{\mathcal {O}(\gamma )}$. The main idea lies in the extension of Reduction Rule 2 by considering the notion of r-twins. Recall that, intuitively, two vertex sets A and B are r-twins if their vertices have the same neighborhoods outside of A and B (for a formal definition, see Section 2).

Reduction Rule 5

Let $T_1, \ldots , T_x\subseteq V(G)$ be a set of x many r-twins, for some $r\in [\ell ]$. If $x\ge k+\ell +2$ , then remove all vertices in $T_x$ from G.

Lemma 5

Reduction Rule 5 is correct.

Proof

Let $G'{:}{=}G-T_x$ be the reduced graph. Before we prove the correctness, we make the following observation for any connected $\ell$-COC set S of size at most k. Since $\vert S \vert \le k$ and $x\ge k+\ell +2$, we observe that there are at least $\ell +2$ many r-twins, denoted by $T_1, \ldots , T_{\ell +2}$, which contain no vertices of S. If $N(T_i)\not \subseteq S$ for some $i\in [\ell +2]$, then there exists a vertex $z\in V(G)\setminus S$ such that $z\in N(T_i)$ for each $i\in [\ell +2]$ since $T_1, \ldots , T_{\ell +2}$ are r-twins. Since $\ell \ge 1$, we conclude that $V(G)\setminus S$ has a connected component of size at least $r(\ell +2)+1\ge \ell +3$, a contradiction to the fact that S is a connected $\ell$-COC set. Thus, $N(T_i)\subseteq S$.

Hence, if G contains a connected $\ell$-COC set S of size at most k, then the set $S'{:}{=}S\setminus T_x$ is also a connected $\ell$-COC set for G and thus also for $G'$. Conversely, if $G'$ contains a connected $\ell$-COC set $S'$ of size at most k, then by the above argumentation, $S'$ is also a connected $\ell$-COC set of size at most k for G. $\square$

Note that Reduction Rule 5 can be exhaustively performed in polynomial time since the r-twin relation can be computed in $n^{2r+\mathcal {O}(1)}$ time and since $r\in [\ell ]$ is a constant. To obtain the kernel, it is also necessary to remove small connected components.

Reduction Rule 6

If G contains a connected component Z with at most $\ell$ vertices, then delete Z.

The correctness of this rule is obvious. Moreover, the rule can be exhaustively applied in linear time. The following theorem, which generalizes Theorem 4, is now a direct consequence of exhaustively applying Reduction Rules 5 and 6 and the subsequent application of Lemma 2.

Theorem 5

Connected $\ell$-COC has a kernel of size $k^{\mathcal {O}(\gamma )}$ for constant $\ell$.

3.4 Induced Matching

In this section, we provide a kernel of size $(\gamma k)^{\mathcal {O}(\gamma )}$ for Induced Matching :

Induced Matching is W[1]-hard for the parameter k on general graphs. For c-closed graphs, we developed a kernel with $\mathcal {O}\left( c^7 k^8\right)$ vertices [21]. For d-degenerate graphs, Kanj et al. [18] and Erman et al. [13] independently presented kernels of size $k^{\mathcal {O}(d)}$. Later, Cygan et al. [5] provided a matching lower bound $k^{o(d)}$ on the kernel size. Note that this also implies the nonexistence of $k^{o(\gamma )}$-size kernels unless coNP $\subseteq$ NP/poly.

It turns out that Lemma 1 is again helpful in designing a $k^{\mathcal {O}(\gamma )}$-size kernel for Induced Matching . In a nutshell, we show that the application of a series of reduction rules results in a graph with a $\left( \gamma k\right) ^{\mathcal {O}(1)}$-size vertex cover. We do so by combining the kernelization of Erman et al. [13] for degenerate graphs with our previous one for c-closed graphs [21]. Lemma 1 and the reduction rule which removes twin vertices then give us a kernel of size $(\gamma k)^{\mathcal {O}(\gamma )}$.

Erman et al. [13] use the following observation for degenerate graphs.

Lemma 6

([13, Proof of Theorem 2.10]) Any d-degenerate graph G with a matching M has an induced matching of size $\vert M \vert / (4d + 1)$.

Ideally, we would like to prove a lemma analogous to Lemma 6 on weakly $\gamma$-closed graphs. Note, however, that a complete graph on n vertices (which is weakly 1-closed) has no induced matching of size 2, although it contains a matching of size $\lfloor n / 2 \rfloor$. So we follow a different route, and prove an analogous lemma on weakly $\gamma$-closed bipartite graphs (there exist bipartite 2-closed graphs whose degeneracy is unbounded; see e.g. Eschen et al. [14]). As we shall see, this serves our purposes.

Lemma 7

Suppose that G is a weakly $\gamma$-closed bipartite graph with a bipartition (A, B). If G has a matching M of size $f_\gamma (k) {:}{=}4 \gamma k^2 + 3k$, then G has an induced matching of size k.

Proof

Recall that $Q^\sigma (v) {:}{=}\{ u \in N(v) \mid u \text { appears after }v\text { in }\sigma \}$. Let $S \subseteq V(G)$ be the set of vertices v such that $\vert Q(v) \vert \ge \gamma k$. Suppose that $\vert S \vert \ge 2k$. Then, we may assume that $\vert A \cap S \vert \ge k$. Let $A' \subseteq A \cap S$ be an arbitrary vertex set of size exactly k and consider some vertex $v \in A'$. Since $\vert Q(v) \cap N(v') \vert < \gamma$ for every $v' \in A' \setminus \{ v \}$ by Observation 1, we have $\vert Q(v) \setminus \bigcup _{v' \in A' \setminus \{ v \}} N(v') \vert > 0$ for each $v\in A'$. Consequently, there is at least one vertex $q_v\in Q(v) \setminus \bigcup _{v' \in A' \setminus \{ v \}} N(v')$. Then, the edge set $\{ v q_v \mid v \in A' \}$ forms an induced matching of size k in G.

Now, suppose that $\vert S \vert < 2k$. By the definition of S, it holds that $\vert Q_{G - S}(v) \vert \le \vert Q_G(v) \vert \le \gamma k$ for each vertex $v \in V(G) \setminus S$. Hence, $G - S$ has degeneracy at most $\gamma k$. Since $G-S$ has a matching $M_{G - S}$ of size at least $\vert M \vert - \vert S \vert \ge f_\gamma (k)-2k= 4 \gamma k^2 + k$, Lemma 6 yields an induced matching of size at least $\left( 4 \gamma k^2 + k\right) / (4\cdot \gamma k + 1) \ge k$. $\square$

We use the following reduction rule to sparsify the graph G so that every sufficiently large vertex set contains a large independent set (see Lemma 9).

Reduction Rule 7

If for some vertex $v \in V(G)$, there is a maximum matching $M_v$ of size at least $2 \gamma k$ in G[Q(v)], then delete v.

Lemma 8

Reduction Rule 7 is correct.

Proof

Let $G' {:}{=}G - v$. Suppose that G has an induced matching M of size k. If $v \notin V(M)$, then M is also an induced matching in $G'$. So assume that $vv' \in M$ for some vertex $v'$. Then, we have $\vert N(u) \cap Q(v) \vert < \gamma$ for any vertex $u \in V\left( M \setminus \{ vv' \}\right)$ by Observation 1. Since $\vert V\left( M^*\right) \vert =2\vert M^*\vert$ for every matching $M^*$ and since $M_v$ is a maximum matching in G[Q(v)] we obtain that $\vert V(M_v) \setminus \bigcup _{u \in V\left( M \setminus \{ vv'\}\right) } N(u) \vert \ge 2 \vert M_v \vert - (\gamma - 1) (2k - 2) > \vert M_v \vert$. By the pigeon-hole principle, this implies that there is an edge $e \in M_v$ such that e is not incident with any vertex in V(M) and no endpoint of e is adjacent to any vertex in $V(M\setminus \{vv'\})$. Then, $\left( M \setminus \{ vv' \}\right) \cup \{ e \}$ is an induced matching of size k in $G'$. $\square$

Lemma 9

Let G be a weakly $\gamma$-closed graph to which Reduction Rule 7 has been exhaustively applied. Then, every vertex set $S \subseteq V(G)$ of size at least $g_\gamma (k) {:}{=}4 \gamma k^2 + k^2$ contains an independent set $I \subseteq S$ of size k.

Proof

Suppose that there is no independent set of size k in $G' {:}{=}G[S]$ for some vertex set S of size $g_\gamma (k)$. For every vertex $v \in S$, let $M_v$ be a maximum matching in $Q_{G'}(v)$ and let $I_v {:}{=}Q_{G'}(v) \setminus V(M_v)$. By Reduction Rule 7, we have $\vert V(M_v) \vert = 2 \vert M_v \vert \le 4\gamma k$. Since $I_v$ is an independent set, we then have $\vert Q_{G'}(v) \vert = \vert M_v \vert + \vert I_v \vert < 4 \gamma k + k$ for every vertex $v \in S$, and thus the degeneracy of $G'$ is smaller than $4 \gamma k + k$. Thus, $G'$ has an independent set of size at least $\vert S \vert / \left( \left( 4 \gamma k + k -1\right) + 1\right) \ge k$, which is a contradiction. $\square$

To identify a part of the graph with a sufficiently large induced matching, we rely on the LP relaxation of Vertex Cover, following our approach [21] to obtain a polynomial kernel on c-closed graphs. Recall that Vertex Cover can be formulated as an integer linear program as follows, using a variable $x_v$ for each $v \in V(G)$:

$$\begin{aligned} \min \sum _{v \in V(G)} x_v \qquad \text {subject to} \quad&x_u + x_v \ge 1 \quad \forall uv \in E(G), \\&x_v \in \{ 0, 1 \} \quad \forall v \in V(G). \end{aligned}$$

We will refer to the LP relaxation of Vertex Cover as VCLP. We use the well-known facts that VCLP always admits an optimal solution in which $x_v \in \{ 0, 1/2, 1 \}$ for each $v \in V(G)$ and that such a solution can be found in polynomial time. Suppose that we have such an optimal solution $(x_v)_{v \in V(G)}$. Let $V_0 {:}{=}\{ v \in V(G) \mid x_v = 0 \}$, $V_1{:}{=}\{ v \in V(G) \mid x_v = 1 \}$, and $V_{1/2} {:}{=}\{ v \in V(G) \mid x_v = 1 / 2 \}$. Also, let ${{\,\mathrm{\mathsf opt}\,}}(G)$ be the optimum of VCLP. We show that we can immediately return Yes, whenever ${{\,\mathrm{\mathsf opt}\,}}(G)$ is sufficiently large:

Reduction Rule 8

If ${{\,\mathrm{\mathsf opt}\,}}(G) \ge 2 g_\gamma \left( g_\gamma \left( f_\gamma (k)\right) \right)$, then return Yes.

Here, the functions $f_\gamma$ and $g_\gamma$ are as specified in Lemmas 7 and 9, respectively.

Lemma 10

Reduction Rule 8 is correct.

Proof

We show that G has an induced matching of size k whenever ${{\,\mathrm{\mathsf opt}\,}}(G) \ge 2 g_\gamma \left( g_\gamma \left( f_\gamma (k)\right) \right)$. Let M be an arbitrary maximal matching in G. Since V(M) is a vertex cover, we have ${{\,\mathrm{\mathsf opt}\,}}(G) \le \vert V(M) \vert = 2 \vert M \vert$, and hence $\vert M \vert \ge {{\,\mathrm{\mathsf opt}\,}}(G) / 2 \ge g_\gamma \left( g_\gamma \left( f_\gamma (k)\right) \right)$. Let $M {:}{=}\left\{ a_1 b_1, \dots , a_{ \vert M \vert } b_{ \vert M \vert } \right\}$ and let $A {:}{=}\left\{ a_1, \dots , a_{ \vert M \vert } \right\}$ and $B {:}{=}\left\{ b_1, \dots , b_{ \vert M \vert } \right\}$. By Lemma 9, there exists an independent set $A' \subseteq A$ of size $s' {:}{=}g_{\gamma }\left( f_\gamma (k)\right)$. Without loss of generality, suppose that $A' = \left\{ a'_1, \dots , a'_{s'} \right\}$ and let $B' {:}{=}\left\{ b'_1, \dots , b'_{s'} \right\}$ be the set of vertices matched to $A'$ in M. Again by Lemma 9, we obtain an independent set $B'' \subseteq B'$ of size $s'' {:}{=}f_\gamma (k)$. We assume without loss of generality that $B'' = \left\{ b''_1, \dots , b''_{s''} \right\}$. Let $A'' {:}{=}\left\{ a''_1, \dots , a''_{s''} \right\}$ be the set of vertices matched to $B''$ in M. Then, $G\left[ A'' \cup B''\right]$ is a bipartite graph with a matching of size at least $s'' = f_\gamma (k)$. By Lemma 7, $G\left[ A'' \cup B''\right]$ has an induced matching of size k. $\square$

Since ${{\,\mathrm{\mathsf opt}\,}}(G) = \vert V_{1/2} \vert / 2 + \vert V_1 \vert$, it holds that $\vert V_{1/2} \vert / 2 + \vert V_1 \vert \le 2 g_\gamma \left( g_\gamma \left( f_\gamma (k)\right) \right) \in \mathcal {O}\left( \gamma ^7 k^8\right)$ after the application of Reduction Rule 8. Hence, it remains to bound the size of $V_0$. To do so, it suffices to remove twins:

Reduction Rule 9

If $N(u) = N(v)$ for some vertices $u, v \in V(G)$, then delete v.

Since an induced matching contains at most one of u and v, the rule is obviously correct. We are finally ready to utilize Lemma 1 to derive an upper bound on $V_0$: Since $V_0$ is an independent set, Lemma 1 gives us $\vert V_0 \vert \in \left| V_{1/2} \cup V_{1} \right| ^{\mathcal {O}(\gamma )} \in (\gamma k)^{\mathcal {O}(\gamma )}$. Thus, we have the following result.

Theorem 6

Induced Matching has a kernel of size $(\gamma k)^{\mathcal {O}(\gamma )}$.

4 Independent Set and Ramsey-Type Problems

We now investigate the kernelization complexity of Independent Set, where we are given a graph G and an integer k, and ask whether G has an independent set of size k. Independent Set admits a kernel with $\mathcal {O}\left( ck^2\right)$ vertices and $\mathcal {O}\left( c^2 k^3\right)$ edges [21] and a kernel with $\mathcal {O}\left( \gamma k^2\right)$ and $\mathcal {O}\left( \gamma ^2 k^3\right)$ edges [20]. We show a lower bound for these parameterizations: unless coNP $\subseteq$ NP/poly, Independent Set admits no kernel of bitsize $k^{2-\epsilon }$ and no kernel with $k^{4/3 - \varepsilon }$ vertices even if the c-closure is constant. We also show a kernel lower bound of size $c^{1 - \varepsilon } k^{\mathcal {O}(1)}$ for any $\varepsilon$. We also consider the following related problem where $\mathcal {G}$ is a hereditary graph class containing all complete graphs and edgeless graphs.

Khot and Raman [19] showed that $\mathcal {G}$-Subgraph is FPT when parameterized by k, using Ramsey’s theorem: for any $k \in \mathbb {N}$, any graph G on at least $R(k)\in 2^{\mathcal {O}(k)}$ vertices contains a clique of size k or an independent set of size k. Ramsey , the special case where $\mathcal {G}$ is the family of all complete and edgeless graphs, admits no polynomial kernel unless coNP $\subseteq$ NP/poly [22]. For several further graph classes $\mathcal {G}$, such as the cluster graphs, it is also known that $\mathcal {G}$-Subgraph admits no polynomial kernel [24]. Our contribution for $\mathcal {G}$-Subgraph is two-fold: First, we observe that the lower bounds for Independent Set on graphs with constant c-closure also hold for Ramsey . This complements a kernel for $\mathcal {G}$-Subgraph with $\mathcal {O}\left( c k^2\right)$ vertices [21].² Second, we provide a kernel of size $k^{\mathcal {O}(\gamma )}$. To show our kernel lower bounds, we will use weak q-compositions. Weak q-compositions exclude kernels of size $\mathcal {O}\left( k^{q - \varepsilon }\right)$ for $\varepsilon > 0$.

Definition 4

([7, 17]) Let $q\ge 1$ be an integer, let $L_1\subseteq \{0,1\}^*$ be a decision problem, and let $L_2\subseteq \{0,1\}^*\times \mathbb {N}$ be a parameterized problem. A weak q-composition from $L_1$ to $L_2$ is a polynomial-time algorithm that on input $x_1, \ldots ,x_{t^q}\in \{0,1\}^n$ outputs an instance $\left( y,k'\right) \in \{0,1\}^*\times \mathbb {N}$ such that:

• $(y,k')\in L_2 \Leftrightarrow x_i\in L_1 \text { for some } i \in [t^q]$, and

• $k'\le t\cdot n^{\mathcal {O}(1)}$.

Lemma 11

([4, 7, 17]) Let $q\ge 1$ be an integer, let $L_1\subseteq \{0,1\}^*$ be an NP-hard problem, and let $L_2\subseteq \{0,1\}^*\times \mathbb {N}$ be a parameterized problem. If there is a weak q-composition from $L_1$ to $L_2$, then $L_2$ has no compression of size $\mathcal {O}(k^{q-\epsilon })$ for any $\epsilon >0$, unless coNP $\subseteq$ NP/poly.

Weak Composition We give a weak composition from the following problem:

A standard reduction from a restricted variant of 3-SAT (for instance, each literal appears exactly twice [2]) shows that Multicolored Independent Set is NP-hard even when $\Delta _G \in \mathcal {O}(1)$ and $\vert V_i \vert \in \mathcal {O}(1)$ for all $i \in [k]$. Let $[t]^q$ be the set of q-dimensional vectors whose entries are in [t]. Suppose that $q\ge 2$ is a constant and that we are given $t^q$ instances $\mathcal {I}_{x} = \left( G_{x}, \left( V_x^1, \dots , V^k_x\right) \right)$ for $x \in [t]^q$, where $\Delta _{G_x} \in \mathcal {O}(1)$ and $\left| V^i_x \right| \in \mathcal {O}(1)$ for all $x \in [t]^q$ and $i \in [k]$. We construct an Independent Set instance $\left( H, k'\right)$. The kernel lower bound of size $k^{2-\epsilon }$ will be based on the special case $q=2$. To obtain the lower bound of $c^{1-\epsilon }k^{\mathcal {O}(1)}$, however, we need the composition to work for all $q\in \mathbb {N}$. Hence, we give a generic description in the following. First, we construct a graph $H_i$ as follows for every $i \in [k]$ (see Fig. 1 for an illustration):

• For every $x \in [t]^q$, include $V^i_x$ into $V(H_i)$.

• For every $r \in [q]$, introduce a path $P^i_r$ on $2t - 2$ vertices. We label the $(2j - 1)$-th vertex as $p^{i}_{r, j, 1}$ and the 2j-th vertex as $p^{i}_{r, j + 1, 2}$ (see Fig. 2 for an illustration). Note that $V\left( P^i_r\right) = \left\{ p^i_{r, j, 1}, p^i_{r, j + 1, 2} \mid j \in [t - 1] \right\}$. For every $j \in [t]$, we new define the set $P^i_{r, j}$: let $P^{i}_{r, 1} = \left\{ p^i_{r, 1, 1} \right\}$, $P^i_{r, t} = \left\{ p^i_{r, t, 2} \right\}$, and $P^i_{r, j} =\left\{ p^i_{r, j, 1}, p^i_{r, j, 2} \right\}$ for $j \in [2, t - 2]$.

• For every $r \in [q]$ and $j \in [t]$, add edges such that $P^i_{r, j} \cup \bigcup _{x \in [t]^q, x_r = j} V_x^i$ forms a clique.

Now, construct H by taking the disjoint union of the $H_i$, $i \in [k]$, and adding the following:

• For every $x \in [t]^q$, add edges such that $H[V(G_x)] = G_x$.

• For every $i \in [k - 1]$, $r \in [q]$, and $j \in [t - 1]$, add edges $p^i_{r, j, 1} p^{i + 1}_{r, j + 1, 2}$ and $p^{i + 1}_{r, j, 1} p^{i}_{r, j + 1, 2}$.

This concludes the construction of H. Let $k' {:}{=}qkt - qk + k$.

We call the vertices of $\bigcup _{x \in [t]^q, i \in [k]} V^i_x$ the instance vertices. The other vertices, which are on $P^i_r$ for some $i \in [k]$ and $r \in [q]$, serve as instance selectors: As we shall see later, any independent set J of size $k'$ in H contains exactly $t - 1$ vertices of $P^i_r$ for every $i \in [k]$ and $r \in [q]$. In fact, there is exactly one $j \in [t]$ such that $J \cap P^i_{r, j} = \emptyset$ and $\left| J \cap P^i_{r, j'} \right| = 1$ for all $j' \in [t] \setminus \{ j \}$. Consequently, J contains no instance vertex in $V^i_x$ for $x_r \ne j$, and thereby, j is selected for the r-th dimension. Next, we bound the c-closure of H and prove the correctness of the reduction.

Fig. 1

An illustration of $H_i$ for $q=2$ and $t=3$. Each thick red box corresponds to a clique

Fig. 2

An illustration of $P^i_r$ and $P^{i + 1}_r$ for $t = 5$

Lemma 12

It holds that ${{\,\mathrm{\mathsf cl}\,}}_H \in \mathcal {O}\left( t^{q - 2}\right)$.

Proof

We first show that ${{\,\mathrm{\mathsf cl}\,}}_H\left( v^i_x\right) \in \mathcal {O}\left( t^{q - 2}\right)$ for every instance vertex $v^i_x \in V^i_x$. More precisely, we show that $\vert N(v) \cap N\left( v^i_x\right) \vert \in \mathcal {O}\left( t^{q - 2}\right)$ for every vertex $v \in V(H) \setminus N\left( v^i_x\right)$. By construction, it holds that

$$\begin{aligned} N\left( v^i_x\right) \subseteq U_i^x \cup V(G_x) \cup \bigcup _{r \in [q]} P^i_{r, x_r},\, \text {where } U_i^x = \bigcup _{x' \in \mathcal {X}^q_t, \, \exists r \in [q] :x_r = x_r'} V_i^{x'}. \end{aligned}$$

Since $\left| P^i_{r, x_r} \right| \le 2$ for each $r \in [q]$, we have $\left| N(v) \cap \bigcup _{r \in [q]} P^i_{r, x_r}\right| \le 2q \in \mathcal {O}(1)$. Moreover, we have $\left| N(v) \cap V\left( G_x\right) \right| \in \mathcal {O}(1)$: If $v \in V\left( G_x\right)$, then $\left| N(v) \cap V\left( G_x\right) \right| \le \Delta _{G_x} \in \mathcal {O}(1)$. Otherwise, $N(v) \cap V\left( G_x\right) = V_{i'}^x$ for some $i' \in [k]$, and hence $\left| N(v) \cap V\left( G_x\right) \right| = \left| V_{i'}^x \right| \in \mathcal {O}(1)$.

Thus, it suffices to show that either $v^i_x \in N(v)$ or $\vert N(v) \cap U_i^x \vert \in \mathcal {O}\left( t^{q - 2}\right)$ holds for every vertex $v \in V(H)$. If $v \in V\left( H_{i'}\right)$ for $i' \in [k] \setminus \{ i \}$, then v has $\mathcal {O}(1)$ neighbors in $U_i^x$. Otherwise, $v \in V_i^r$ or $v \in P^i_{r, y_r}$ for some $y \in [t]^q$.

• If $x_r = y_r$ for some $r \in [q]$, then $v^i_x$ and v are adjacent.

• If $x_r \ne y_r$ for all $r \in [q]$, then $N(v) \cap U_i^x \subseteq \bigcup _{z \in \mathcal {Z}} V_i^z$, where $\mathcal {Z}$ is the set of vectors $z \in [t]^q$ such that $x_{r'} = z_{r'}$ and $y_{r''} = z_{r''}$ for some $r', r'' \in [q]$. We claim that $\mathcal {Z}$ contains at most $\mathcal {O}\left( t^{q - 2}\right)$ vectors. As $x_r \ne y_r$ for all $r \in [q]$, we can assume that $r' \ne r''$. So deleting the r-th and the $r'$-th entries from any vector in $\mathcal {Z}$ yields a vector in $[t]^{q - 2}$. Thus, there is a one-to-one correspondence between an element of $\mathcal {Z}$ and $r' \ne r'' \in [q]$, and a vector $z' \in [t]^{q - 2}$. It follows that $\left| \mathcal {Z} \right| \le q(q - 1) t^{q - 2} \in \mathcal {O}\left( t^{q - 2}\right)$, and consequently, $\vert N(v) \cap U_i^x \vert \le \mathcal {O}\left( \left| \mathcal {Z} \right| \right) = \mathcal {O}\left( t^{q - 2}\right)$.

It remains to show that for every pair $v, v'$ of instance selector vertices, we have either $vv'\in E(H)$ or $\vert N(v) \cap N\left( v'\right) \vert \in \mathcal {O}\left( t^{q - 2}\right)$. Suppose that $v \in P^i_{j, r}$ and $v \in P^{i'}_{j', r'}$. If $i \ne i'$, then v and $v'$ have a constant number of neighbors in common. So assume that $i = i'$. If $r = r'$, then either $vv' \in E(H)$ or $N(v) \cap N\left( v'\right) = \emptyset$ holds. Otherwise, it holds that $N(v) \cap N\left( v'\right) \subseteq \bigcup _{z \in \mathcal {Z}} V_i^z$ where $\mathcal {Z}$ is the set of vectors z such that $z_r = j$ and $z_{r'} = j'$. It follows from an analogous argument as above that $\vert \mathcal {Z} \vert = t^{q - 2}$. Since $\left| V^i_x \right| \in \mathcal {O}(1)$ for every $i \in [k]$ and $x \in [t]^q$, we have $\left| N(v) \cap N\left( v'\right) \right| \in \mathcal {O}\left( t^{q - 2}\right)$. $\square$

Lemma 13

The graph $G_x$ has a multicolored independent set of size k for some $x\in [t]^q$ if and only if the graph H has an independent set I of size $k'$.

Proof

Assume that $G_x$ has a multicolored independent set I of size k for some $x \in [t]^q$. One can easily verify that the set J consisting of the following vertices is an independent set in H.

• All vertices of I.

• The vertex $p^i_{r, j, 1}$ for each $r \in [q]$, $i \in [k]$, and $j \in \left[ x_r - 1\right]$.

• The vertex $p^i_{r, j, 2}$ for each $r \in [q]$, $i \in [k]$, and $j \in [x_r + 1, t]$.

The size of J is then $k + \sum _{ r \in [q], i \in [k]} \left( x_r - 1\right) + \left( t - x_r\right) = k + qk(t - 1) = k'$.

Conversely, assume that H has an independent set J of size $k'$. First, we show that $\left| J \cap \bigcup _{x \in [t]^q} V^i_x \right| = 1$ and $\left| J \cap V\left( P^i_r\right) \right| = t - 1$ for every $i \in [k]$ and $r \in [q]$. Let $J^i {:}{=}J \cap V(H_i)$ for each $i \in [k]$. Since $P^i_r$ is a path on $2t - 2$ vertices, $J^i$ contains at most $t - 1$ vertices of $P^i_r$ for every $r \in [q - 1]$, that is, $\left| J^i \cap V\left( P^i_r\right) \right| \le t - 1$. The set of remaining vertices in $H_i$, that is, $V\left( H_i\right) \setminus \bigcup _{r \in [q - 1]} V\left( P^i_r\right)$, can be partitioned into t cliques: $P^i_{q, j} \cup \bigcup _{x \in [t]^q, x_q = j} V_x^i$ for $j \in [t]$. So we have $\left| J^i \cap \left( P^i_{q, j} \cup \bigcup _{x \in [t]^q, x_q = j} V_x^i\right) \right| \le 1$ for each $j\in [t]$. Consequently, we obtain

$$\begin{aligned} \vert J \vert&= \sum _{i \in [k], r \in [q - 1]} \left| J^i \cap P^i_r \right| + \sum _{i \in [k], j \in [t]} \vert J^i \cap \left( P^i_{q, j} \cup \bigcup _{x \in [t]^q, x_q = j} V_x^i\right) \vert \\&\le k(q - 1)(t - 1) + kt = qkt - qk + k = k'. \end{aligned}$$

In fact, the equality holds. Hence, $\left| J^i \cap P^i_r \right| = t - 1$ and $\vert J^i \cap \left( P^i_{q, j} \cup \bigcup _{x \in [t]^q, x_q = j} V_x^i\right) \vert = 1$ for each $i \in [k]$ and $r \in [q - 1]$. An analogous argument (in which $J^i$ is partitioned into $J^i \cap P^i_r$ for $r \in [2, q]$ and $J^i \cap \left( P^i_{1,j} \bigcup _{x \in [t]^q, x_r = j} V_x^i\right)$) also shows that $\left| J^i \cap P^i_q \right| = t - 1$. We thus have for each $i \in [k]$,

$$\begin{aligned}&\left| J^i \cap \bigcup _{x \in [t]^q, x_r = j} V_x^i \right| = \left| J^i \cap \left( \bigcup _{j \in [t]} \left( P^i_{q, j} \cup \bigcup _{x \in [t]^q, x_r = j} V_x^i\right) \right) \right| \\ {}&\quad - \left| J^i \cap P^i_q \right| = t - (t - 1) = 1. \end{aligned}$$

Now, let $x^i \in [t]^q$ be such that $J \cap V_i^{x^i} \ne \emptyset$ for each $i \in [k]$. We claim that $J \cap V(P^i_r) = \left\{ p^i_{r, j, 1} \mid j \in \left[ x_r^i - 1\right] \right\} \cup \left\{ p^i_{r, j, 2} \mid j \in [x_r^i + 1, t]\right\}$ for every $i \in [k]$ and $r \in [q]$:

• If $x_r^i = 1$, then $J \cap V\left( P^i_r\right)$ lies inside the path $P^i_r - \left\{ p^i_{r, 1, 1} \right\}$ on $2t - 3$ vertices, in which there is exactly one independent set of size $t - 1$, namely, $\left\{ p^i_{r, j, 2} \mid j \in [2, t] \right\}$. It follows that $J \cap V\left( P^i_r\right) = \left\{ p^i_{r, j, 2} \mid j \in [2, t] \right\}$.

• If $x_r^i = t$, then $J \cap V\left( P^i_r\right)$ lies inside the path $P^i_r - \{ p^i_{r, t, 2} \}$ on $2t - 3$ vertices, in which there is exactly one independent set of size $t - 1$, namely, $\left\{ p^i_{r, j, 1} \mid j \in [t - 1] \right\}$. It follows that $J \cap V\left( P^i_r\right) = \left\{ p^i_{r, j, 1} \mid j \in [t - 1] \right\}$.

• If $x_r^i \in [2, t - 1]$, then $J \cap V\left( P^i_r\right)$ lies in the disjoint union of $P^i_r\left[ \bigcup _{j \in \left[ x_r^i - 1\right] } P^i_{r, j}\right]$ and $P^i_r\left[ \bigcup _{j \in \left[ x_r^i + 1, t\right] } P^i_{r, j}\right]$. Since $P^i_r\left[ \bigcup _{j \in \left[ x_r^i - 1\right] } P^i_{r, j}\right]$ (and $P^i_r\left[ \bigcup _{j \in \left[ x_r^i + 1, t\right] } P^i_{r, j}\right]$) is a path on $2x_r^i - 3$ (and $2t - 2x_r^i - 1$, respectively) vertices, it has only one independent set of size $x_r^i - 1$ (and $t - x_r^i$, respectively): $\left\{ p^i_{r, j, 1} \mid j \in \left[ x_r - 1\right] \right\}$ (and $\left\{ p^i_{r, j, 2} \mid j \in \left[ x_r + 1, t\right] \right\}$, respectively) Thus, our claim holds.

Finally, we show that $x^i = x^{i + 1}$ for all $i \in [k - 1]$. Assume to the contrary that $x_r^i \ne x^{i + 1}_r$ for some $i \in [k - 1]$ and $r \in [q]$. If $x_r^i < x_r^{i + 1}$ ($x_r^i > x_{r}^{i + 1}$), then J contains $p^i_{r, x_r^i + 1, 2}$ and $p^{i + 1}_{r, x_r^i, 1}$ ($p^i_{r, x_{r}^{i + 1}, 1}$ and $p^{i + 1}_{r, x_{r}^{i + 1} + 1, 2}$, respectively). By construction, these vertices are adjacent in H, which contradicts the fact that J is an independent set. We thus have shown that $x^i = x^{i + 1}$ for all $i \in [k - 1]$, and thereby, there exists $x \in [t]^q$ such that $J \cap V^i_x \ne \emptyset$ for all $i \in [k]$. Since $H\left[ \bigcup _{i \in [k]} V^i_x\right] = G_x$, it follows that $J \cap \bigcup _{i \in [k]} V^i_x$ is a multicolored independent set of size k in $G_x$. $\square$

For $q = 2$, we have a weak 2-composition from Multicolored Independent Set to Independent Set on $\mathcal {O}(t^{q - 2}) = \mathcal {O}(1)$-closed graphs by Lemmas 12 and 13. Since the constructed graph H has no clique of size $k'$, the construction also constitutes a weak 2-composition to Ramsey on $\mathcal {O}(1)$-closed graphs. Thus, Lemma 11 implies the following:

Theorem 7

For any $\varepsilon > 0$, neither Independent Set nor Ramsey has a kernel of bitsize $k^{2 - \varepsilon }$ on graphs of constant c-closure, unless coNP $\subseteq$ NP/poly.

By Theorem 7, neither Independent Set nor Ramsey admit a kernel with $k^{1 - \varepsilon }$ vertices. We improve this bound on the number of vertices, taking advantage of the fact that any n-vertex c-closed graph can be encoded using $\mathcal {O}(c n^{1.5} \log n)$ bits in polynomial time [14]. Assume for a contradiction that Independent Set or Ramsey admit a kernel with $k^{4/3 - \varepsilon '}$ vertices for constant c. Using the above-mentioned encoding, we obtain a string with $\mathcal {O}\left( k^{(4/3 - \varepsilon ')1.5}\log k\right) = \mathcal {O}\left( k^{2 - \varepsilon }\right)$ bits. So a kernel of $k^{4/3 - \varepsilon '}$ vertices implies that there is a compression of Independent Set or Ramsey with bitsize $\mathcal {O}\bigg (k^{2 - \varepsilon }\bigg )$, a contradiction. Thus, we have the following:

Theorem 8

For any $\varepsilon > 0$, neither Independent Set nor Ramsey has a compression with $k^{4/3 - \varepsilon }$ vertices on graphs of constant c-closure, unless coNP $\subseteq$ NP/poly.

We also obtain another kernel lower bound for Independent Set ; this bound excludes the existence of polynomial kernels (in terms of $c + k$) whose dependence on c is sublinear.

Theorem 9

For any $\varepsilon > 0$, Independent Set  has no kernel of size $c^{1 - \varepsilon } k^{\mathcal {O}(1)}$ unless coNP $\subseteq$ NP/poly.

Proof

We show that Independent Set  admits no kernel of size $c^{1 - \varepsilon } k^{i}$ for any fixed $\varepsilon > 0$ and $i > 0$, unless coNP $\subseteq$ NP/poly. Let q be a sufficiently large fixed integer with $\frac{q - \varepsilon - i}{q - 2} > 1 - \varepsilon$ (that is, $q > \frac{i + 3 \varepsilon - 2}{\varepsilon }$). Perform the construction of the weak composition with this value of q. Recall that in the constructed instance we have $c\in \mathcal {O}\left( t^{q-2}\right)$ and $k'{:}{=}qkt-qk+k<qkt$ where $k'$ is the solution size of the constructed Independent Set instance. Now consider the new parameter $\ell {:}{=}c^{\frac{1}{q - 2} \left( 1 - \frac{i}{q - \varepsilon } \right) } k'^{\frac{i}{q - \varepsilon }}$. A straightforward calculation shows that

$$\begin{aligned} \ell = c^{\frac{1}{q - 2} \left( 1 - \frac{i}{q - \varepsilon } \right) } k'^{\frac{i}{q - \varepsilon }}&= \mathcal {O}\left( t^{q-2 \frac{1}{q - 2} \left( 1 - \frac{i}{q - \varepsilon } \right) } (qkt)^{\frac{i}{q - \varepsilon }}\right) \\ {}&= \mathcal {O}\left( t\ (qk)^{\frac{i}{q - \varepsilon }}\right) = t \cdot k^{\mathcal {O}(1)} \end{aligned}$$

and hence the construction can also be viewed as a weak q-decomposition from Multicolored Independent Set to Independent Set  parameterized by $\ell$. Thus, Lemma 11 implies that Independent Set  does not admit a kernel of size $\ell ^{q - \varepsilon } = c^{\frac{q - \varepsilon - i}{q - 2}} k'^i > c^{1 - \varepsilon } k'^i$. $\square$

Finally, we show that $\mathcal {G}$-Subgraph has a kernel of size $k^{\mathcal {O}(\gamma )}$ for any graph class $\mathcal {G}$ containing all complete graphs and empty graphs.

Proposition 2

There exists a function $R_\gamma (a, b) \in (a \cdot b)^{\gamma + \mathcal {O}(1)}$ such that any weakly gamma-closed graph G on at least $R_\gamma (a, b)$ vertices has a clique of size a or an independent set of size b.

Proof

The Ramsey number R(a, b) denotes the smallest number such that every graph on R(a, b) vertices contains a clique of size a or an independent set of size b. It is known that $R(a, b) \le \left( {\begin{array}{c}a + b\\ b\end{array}}\right)$. So whenever $a \le \gamma$ or $b \le \gamma$, we have $R_{\gamma }(a, b) \le \left( {\begin{array}{c}a + b\\ \gamma \end{array}}\right) \in \mathcal {O}((a + b)^{\gamma })$. For $a, b > \gamma$, let $R_\gamma (a, b)$ be some number greater than $ab \left( {\begin{array}{c}b\\ \gamma \end{array}}\right) + b \left( {\begin{array}{c}a+\gamma \\ \gamma \end{array}}\right) b^{\gamma }$.

Let G be a weakly $\gamma$-closed graph on $n = \vert V(G) \vert \ge R_\gamma (a, b)$ vertices. Moreover, let $v_1, \dots , v_n$ be a weak closure ordering $\sigma$ of G. Divide V(G) into b subsets $V_1, \dots , V_b$ of equal size:³ let $V_i = \left\{ v_{((b - i)n / b) + 1}, \dots , v_{(b - i + 1)n / b} \right\}$ for each $i \in [b]$. Notably, $V_1$ is the set of n/b vertices occurring last in $\sigma$ and $V_b$ is the set of n/b vertices occurring first in $\sigma$. Moreover, let $G_i {:}{=}G\left[ \left\{ v_{(b - i)n / b + 1}, \dots , v_n \right\} \right]$ be the subgraph induced by $\bigcup _{i' \in [i]} V_{i'}$ for each $i \in [b]$. Suppose that G contains no clique of size a. We show that G contains an independent set of size b. More precisely, we prove by induction that $G_i$ contains an independent set of size i for each $i \in [b]$.

This clearly holds for $i = 1$. For $i > 1$, assume that there is an independent set I of size $i - 1$ in $G_{i - 1}$ by the induction hypothesis. In the following, we consider subsets X of size at most $\gamma$ of I to obtain an independent set $I'$ of size at least i.

First, consider vertex sets $X \subseteq I$ of size $\gamma$ and $V_X{:}{=}\left\{ v \in V_i \mid N_G(v) \supseteq X\right\}$. Note that $X\subseteq Q(v)$ for each $v\in V_X$. Hence, since G is weakly $\gamma$-closed, $V_X$ is a clique. It follows that $\left| V_X \right| < a$. Therefore, less than $a \left( {\begin{array}{c}b\\ \gamma \end{array}}\right)$ vertices of $V_i$ are adjacent to at least $\gamma$ vertices in I.

Second, consider vertex sets $X \subseteq I$ with $X \ne \emptyset$ and $\vert X \vert < \gamma$. Furthermore, let $V_X' = \left\{ v \in V_i \mid N_G(v) \cap I = X\right\}$. Since $n>R_\gamma (a,b)$, we conclude that $\vert V_i\vert \ge a \left( {\begin{array}{c}b\\ \gamma \end{array}}\right) + \left( {\begin{array}{c}a+\gamma \\ \gamma \end{array}}\right) b^\gamma$. Now since less than $a \left( {\begin{array}{c}b\\ \gamma \end{array}}\right)$ vertices of $V_i$ are adjacent to at least $\gamma$ vertices in I, we observe that at least $\left( {\begin{array}{c}a+\gamma \\ \gamma \end{array}}\right) b^\gamma$ vertices of $V_i$ are adjacent to at most $\gamma -1$ vertices in I. Note that there are less than $(i-1)^{\gamma }<b^{\gamma }$ subsets X of size at most $\gamma -1$ of I.

Hence, there exists $X \subseteq I$ of size at most $\gamma - 1$ such that $\vert V_X' \vert > R(a, \gamma )=\left( {\begin{array}{c}a+\gamma \\ \gamma \end{array}}\right)$. By Ramsey’s theorem, we then find an independent set $I' \subseteq V_X'$ of size $\gamma$ (recall that G has no clique of size a). It follows that $(I \setminus X) \cup I'$ is an independent set of size at least i in $G_i$. $\square$

Now, we directly obtain a kernel for $\mathcal {G}$-Subgraph where $\mathcal {G}$ contains all cliques and all independent sets since each graph on $k^{\mathcal {O}(\gamma )}$ vertices contains either a clique or an independent set of size at least k by the bound on the Ramsey number in weakly closed graphs shown in Proposition 2.

Corollary 1

Let $\mathcal {G}$ be a class of graphs containing all cliques and independent sets. $\mathcal {G}$-Subgraph has a kernel of size $k^{\mathcal {O}(\gamma )}$.

5 Dominating Set in Bipartite and Split Graphs

We now develop a $k^{\mathcal {O}(\gamma )}$-size kernel for Dominating Set on split graphs and a kernel of size $k^{\mathcal {O}(\gamma ^2)}$ on graphs with constant clique size. Dominating Set is defined as follows:

Dominating Set is W[2]-hard when parameterized by k, even on split graphs and bipartite graphs [27] and admits kernels of size $\mathcal {O}\left( k^{\min \{ i^2 \!,\, j^2 \}}\right)$ on $K_{i, j}$-subgraph-free graphs when $\min \{ i, j \}$ is constant [26] and of size $k^{\mathcal {O}(c)}$ on c-closed graphs [21]. These kernels are essentially optimal: unless coNP $\subseteq$ NP/poly, there is no kernel of size $k^{o\left( d^2\right) }$ [5] or $k^{o(c)}$ [21]. Finally, Dominating Set is FPT with respect to $k+\gamma$ but the existence of a polynomial kernel for fixed $\gamma$ is open [25].

A Kernel for Bipartite Graphs First we obtain a polynomial kernel for Dominating Set on weakly $\gamma$-closed bipartite graphs. To do so, we exploit the fact that any weakly closed graph is bipartite-subgraph-free (not necessarily induced), whenever its maximum clique size $\omega$ is fixed:

Lemma 14

Any weakly $\gamma$-closed graph with maximum clique size $\omega$ is $K_{\rho , \rho }$-subgraph-free, where $\rho {:}{=}\gamma + \omega + 1$.

Proof

Assume towards a contradiction that G contains $K_{\rho , \rho }$ as a subgraph. Then, there are disjoint vertex sets $U {:}{=}\left\{ u_1, \dots , u_{\rho } \right\}$ and $W {:}{=}\{ w_1, \dots , w_{\rho } \}$ such that $uw \in E(G)$ for every $u \in U$ and $w \in W$. Without loss of generality, assume that $u_i$ (and $w_i$) appears before $u_{i + 1}$ (and $w_{i + 1}$, respectively) for each $i \in [\rho - 1]$ in a weak closure ordering $\sigma$ of G. We can also assume without loss of generality that $u_{\omega + 1}$ appears after $w_{\omega + 1}$ in $\sigma$. Then, we have $U' {:}{=}\{ u_{\omega + 1}, \dots , u_{\rho } \} \subseteq Q(w_{i})$ for every $i \in [\omega + 1]$, and hence we obtain $\vert Q(w_i) \cap Q\left( w_{i'}\right) \vert \ge \vert U' \vert = \gamma$ for all $i < i' \in [\omega + 1]$. By the definition of the weak closure, $\left\{ w_1, \dots , w_{\omega + 1} \right\}$ is a clique. This contradicts the assumption that G has no clique of size $\omega + 1$. $\square$

For a graph G, let $\omega _G$ denote the maximum clique size of G (we drop the subscript when clear from context). Since Dominating Set admits a kernel of size $k^{\min \{ i^2\!, j^2 \}}$ on $K_{i, j}$-subgraph-free graphs [26], we immediately obtain the following proposition from Lemma 14:

Proposition 3

Dominating Set has a kernel of size $k^{\mathcal {O}\left( (\gamma + \omega )^2\right) }$.

Since $\omega _G \le 2$ for any bipartite graph G, we also obtain the following.

Corollary 2

Dominating Set has a kernel of size $k^{\mathcal {O}\left( \gamma ^2\right) }$ in bipartite graphs.

A Kernel for Split Graphs A graph G is a split if there is a bipartition (C, I) of V(G) such that I is an independent set and C is a clique. Note that such a bipartition can be found in polynomial time. In the following, we fix one such bipartition. Furthermore, we assume that there are no isolated vertices. We start with a simple observation on Dominating Set on split graphs:

Observation 3

If D is a solution of (G, k), then there exists a solution $D'$ such that $D' \cap I = \emptyset$.

We obtain the following reduction rule as a consequence of Observation 3.

Reduction Rule 10

If there is a vertex $v \in I$ such that $N(v) = C$, then delete v.

We will assume that Reduction Rule 10 is applied exhaustively. Afterwards, we may assume that, for the fixed bipartition (C, I) of V(G), the clique C is a maximum clique. We will show that there is a weak closure ordering $\sigma$ in which every vertex in C appears before any vertex in I (Lemma 16). Before doing so, we prove an auxiliary lemma on the relation of the minimum degree $\delta (G)$ and the weak closure in reduced split graphs with maximum clique C.

Lemma 15

If $\delta (G) \ge \gamma$, then there is a vertex $v \in C$ with ${{\,\mathrm{\mathsf cl}\,}}(v) < \gamma$.

Proof

By the definition of the weak $\gamma$-closure, there exists a vertex $v \in V(G)$ such that ${{\,\mathrm{\mathsf cl}\,}}(v) < \gamma$. Clearly, the lemma holds for $v \in C$. So assume that $v \in I$. If there is a vertex $u \in C \setminus N(v)$, then u and v have

$$\begin{aligned} \vert N(u) \cap N(v) \vert \ge \vert N(u) \cap C \vert = \deg (v) \ge \delta (G) \ge \gamma \end{aligned}$$

common neighbors, contradicting the fact that ${{\,\mathrm{\mathsf cl}\,}}(v) < \gamma$. Thus, we have $C \setminus N(v) = \emptyset$. It follows that $C \cup \{ v \}$ is a clique. However, this contradicts the maximality of C, and hence $v \in C$. $\square$

Fig. 3

The ordering $\{a,c,e,g,b,d,f\}$ is a good weak closure ordering of this graph. Furthermore, $s_b=2$, $s_d=1$, and $s_f=0$. Also, $S(b)=\emptyset$, $S(d)=\{e\}$, and $S(f)=\{c,g\}$

We say that a weak closure ordering is good if every vertex in C appears before any vertex in I. An example of a split graph and a corresponding good weak closure ordering is given in Fig. 3. We show the existence of a good weak closure ordering.

Lemma 16

There is a good weak closure ordering.

Proof

We prove the existence of a good closure ordering by induction on the number n of vertices. The lemma clearly holds when $\Delta \le \gamma - 1$, as any ordering of V(G) is a weak closure ordering. So assume that $\Delta \ge \gamma$. We consider two cases: $\delta \ge \gamma$ and $\delta \le \gamma - 1$.

If $\delta \ge \gamma$, then there exists a vertex $v \in C$ such that ${{\,\mathrm{\mathsf cl}\,}}(v) < \gamma$ by Lemma 15. By induction hypothesis, there is a good weak closure ordering $\sigma '$ of $G - v$. Prepending v to $\sigma '$, we obtain a good weak closure ordering of G.

Now suppose that $\delta \le \gamma - 1$. We may assume that there is a vertex $v\in I$ such that $\deg (v)\le \gamma -1$: if only vertices $u\in C$ have $\deg (u)\le \gamma -1$, then $\vert C\vert \le \gamma -1$ and thus for all $v\in I$ we have $\deg (v)\le \gamma -1$. Appending v to a weak closure ordering $\sigma '$ of $G - v$ yields a good weak closure ordering. $\square$

Note that the proof of Lemma 16 is constructive. In fact, one can find a good ordering in polynomial time. Let us fix a good weak closure ordering $\sigma = v_1, \dots , v_n$. Observe that $N(v) = P(v)$ for every vertex $v \in I$ with respect to $\sigma$.

With a good weak closure ordering at hand, we now develop further reduction rules for Dominating Set in split graphs. To do so, we use the notion of sunflowers. Recall that a sunflower with a core T is a set family $\mathcal {S} = \left\{ S_1, \dots , S_k \right\}$ such that $S_i \cap S_j = T$ for all $i < j \in [k]$. We say that the sets $S_i \setminus T$ are the petals of $\mathcal {S}$.

Lemma 17

([11, sunflower lemma]) Let $\mathcal {F}$ be a family of sets, each of size at most $\lambda$. If $\vert \mathcal {F} \vert \ge \lambda ! k^\lambda$, then $\mathcal {F}$ contains a sunflower with at least k petals. Moreover, such a sunflower can be found in polynomial time.

The sunflower lemma requires the cardinality of sets to be bounded. Since the vertices of G could have arbitrary many neighbors, this does not seemingly help. However, by crucially exploiting the weak $\gamma$-closure, we will work around this issue.

To do so, let us introduce further notation. Recall that, since $\sigma$ is a good weak closure ordering, we have $C = \left\{ v_1, \dots , v_{ \vert C \vert } \right\}$ and $I = \left\{ v_{ \vert C \vert + 1}, \dots , v_{n} \right\}$. For each vertex $u \in I$, let $s_u$ be the smallest $i \in [0, \vert C \vert - 1]$ such that $v_{i + 1} \notin N(u)$ and let $S(u) {:}{=}N(u) \setminus \left\{ v_1, \dots , v_{s_u} \right\}$. An example is given in Fig. 3. Note that Reduction Rule 10 ensures that such an integer $s_u$ always exists.

Lemma 18

For each vertex $u \in I$, it holds that $\vert S(u) \vert \le \gamma - 1$.

Proof

By the definition of $s_u$, there is no edge between u and $v_{s_u + 1}$. Since C is a clique and $S_u\subseteq C$, we conclude that $v_{s_u+1}$ is adjacent to all vertices in S(u) and thus, we have $\vert S(u) \vert = \left| N(u) \cap Q\left( v_{s_u + 1}\right) \right| \le \gamma - 1$. $\square$

Consequently, it follows from Lemma 17 that the set family $\{ S(v) \mid v \in I \}$ contains a sunflower with at least $k + 2$ petals whenever $\vert I \vert \ge (\gamma - 1)! (k + 2)^{\gamma - 1}$.

Reduction Rule 11

If there is a set $I' \subseteq I$ of $k + 2$ vertices such that $\mathcal {S} {:}{=}\{ S(v) \mid v \in I' \}$ is a sunflower, then delete $u {:}{=}{{\,\mathrm{\mathsf argmax}\,}}_{u' \in I'} s_{u'}$.

Lemma 19

Reduction Rule 11 is correct.

Proof

Let $G' {:}{=}G - u$. Let D be a solution of (G, k). According to Observation 3 we can assume that $D\cap I=\emptyset$. Hence, $u\notin D$ and thus D is also a solution for $(G',k)$.

Conversely, suppose that $D'$ is a solution of $(G', k)$. Let T be the core of $\mathcal {S}$ and let $A {:}{=}\left\{ v_1, \dots , v_{s_u} \right\} \cup T$. We prove by contradiction that $A \cap D' \ne \emptyset$. Recall that $N(u') = S(u') \cup \left\{ v_1, \dots , v_{s_{u'}} \right\}$ for every vertex $u' \in I' \setminus \{ u \}$. Since $s_{u'} < s_u$ (recall that $u = {{\,\mathrm{\mathsf argmax}\,}}_{u' \in I'} s_{u'}$), we have $N(u') \setminus A = S(u') \setminus A \subseteq S(u') \setminus T$. Thus, the sets $N(u') \setminus A$ are the petals of $\mathcal {S}$ and thus they are pairwise disjoint for $u' \in I' \setminus \{ u \}$. If $A \cap D' = \emptyset$, then $D'$ must contain at least one vertex from $N(u') \setminus A$ for every vertex $u' \in I' \setminus \{ u \}$ since $u'$ is not dominated by $D'$. However, this contradicts the fact that $\vert D' \vert \le k$ since at least $k+1$ petals are left, and consequently, we have $A \cap D' \ne \emptyset$. Since $N(u) \supseteq A$, $D'$ also dominates u in G, implying that $D'$ is also a solution of (G, k). $\square$

Reduction Rule 11 gives us an upper bound on the size of I. We use the following reduction rule to obtain an upper bound on the size of C. The correctness follows from Observation 3.

Reduction Rule 12

If there are vertices $u, v \in C$ such that $N(u) \supseteq N(v)$, then delete v.

Finally, we show that the aforementioned reduction rules yield an optimal (up to constants in the exponent) kernel.

Theorem 10

Dominating Set has a kernel of size $(\gamma k)^{\mathcal {O}(\gamma )}$ in split graphs.

Proof

We apply Reduction Rules 10 to 12 exhaustively. First, note that $\vert I \vert \in (\gamma k)^{\mathcal {O}(\gamma )}$, since otherwise Reduction Rule 11 can be further applied by Lemma 17. Thus, it remains to show that $\vert C \vert \in (\gamma k)^{\mathcal {O}(\gamma )}$. By Lemma 18, there are at most $(\gamma - 1) \vert I \vert$ vertices in C that are contained in S(u) for some vertex $u \in I$. Since $(\gamma - 1) \vert I \vert \in (\gamma k)^{\mathcal {O}(\gamma )}$, it remains to show that $\vert C' \vert \in (\gamma k)^{\mathcal {O}(\gamma )}$, where $C' \subseteq C$ be the set of vertices not contained in S(u) for any $u \in I$. Let $Z {:}{=}\left\{ s_u \mid u \in I \} \cup \{ 0, \vert C \vert - 1 \right\}$ and let $0 = z_1< z_2< \ldots < z_{ \vert Z \vert } = \vert C \vert + 1 \in Z$ be the indices in the good weak closure ordering $\sigma$. Next, we show that for each $j\in [\vert Z\vert -1]$ there is at most one vertex $v_i\in C'$ such that $z_j\le i< z_{j+1}$. Assume towards a contradiction, that there are at least two vertices $v_i, v_{i'}\in C'$ such that $z_j\le i,i'<z_{j+1}$. Since neither $v_i$ nor $v_{i'}$ are contained in any $S_u$ for some $u\in I$, we conclude that $N(v_i)\cap I=N\left( v_{i'}\right) \cap I$. Since $v_i,v_{i'}\in C$ we thus obtain that $N(v_i)=N\left( v_{i'}\right)$, which contradicts the fact that Reduction Rule 12 has been exhaustively applied. Thus, we have $\vert C' \vert \le \vert Z \vert - 1\le \vert I \vert + 1 \in (\gamma k)^{\mathcal {O}(\gamma )}$. $\square$

6 Conclusion

We have provided several kernelizations and kernelization lower bounds for classic graph problems on (weakly) closed graphs. How far can our results for Connected Vertex Cover and Capacitated Vertex Cover be extended to other cases of connected or capacitated vertex deletion problems? We did show that Connected $\ell$-COC admits a kernel of size $k^{\mathcal {O}(\gamma )}$. In contrast, Connected Feedback Vertex Set does not admit a polynomial kernel for the solution size k even in 2-closed graphs [6]. Moreover, it seems interesting to extend our polynomial kernelization for Connected Vertex Cover parameterized by $k+c$ to other connected vertex deletion problems. Drawing a borderline between those desired graph properties where connected and capacitated vertex deletion problems do admit a kernel on (weakly) closed graphs and those where they do not would improve our understanding of the circumstances under which (weak) closure can be exploited algorithmically. It is also open whether the Ramsey number of weakly closed graphs can be bounded by $(a+b+\gamma )^{\mathcal {O}(1)}$. Such a bound would immediately improve some of our kernels. Finally, the most important open problem is arguably whether Dominating Set parameterized by the solution size k admits a polynomial kernel on weakly closed graphs. We made partial progress by showing that Dominating Set admits a kernel on weakly closed bipartite graphs and on weakly closed split graphs. Answering this question positively would need further insights into the structure of weakly closed graphs, however. Note that it was recently proven that Dominating Set is fixed-parameter tractable with respect to $k+\gamma$ [25].