Computing the Largest Bond and the Maximum Connected Cut of a Graph

Abstract

The cut-set $\partial (S)$ of a graph $G=(V,E)$ is the set of edges that have one endpoint in $S\subset V$ and the other endpoint in $V\setminus S$, and whenever G[S] is connected, the cut $[S,V\setminus S]$ of G is called a connected cut. A bond of a graph G is an inclusion-wise minimal disconnecting set of G, i.e., bonds are cut-sets that determine cuts $[S,V\setminus S]$ of G such that G[S] and $G[V\setminus S]$ are both connected. Contrasting with a large number of studies related to maximum cuts, there exist very few results regarding the largest bond of general graphs. In this paper, we aim to reduce this gap on the complexity of computing the largest bond, and the maximum connected cut of a graph. Although cuts and bonds are similar, we remark that computing the largest bond and the maximum connected cut of a graph tends to be harder than computing its maximum cut. We show that it does not exist a constant-factor approximation algorithm to compute the largest bond, unless $\text{ P }= \text{ NP }$. Also, we show that Largest Bond and Maximum Connected Cut are NP-hard even for planar bipartite graphs, whereas Maximum Cut is trivial on bipartite graphs and polynomial-time solvable on planar graphs. In addition, we show that Largest Bond and Maximum Connected Cut are NP-hard on split graphs, and restricted to graphs of clique-width w they can not be solved in time $f(w) n^{{o}(w)}$ unless the Exponential Time Hypothesis fails, but they can be solved in time $f(w) n^{{O}(w)}$. Finally, we show that both problems are fixed-parameter tractable when parameterized by the size of the solution, the treewidth, and the twin-cover number.

Appendix

1.1 Faster Algorithms Parameterized by Clique-width (Theorem 10)

In this section, we design faster XP algorithms for both Maximum Connected Cut and Largest Bond when parameterized by clique-width, which run in time $n^{O(w)}$.

Here, we rather use a different graph parameter and its associated decomposition closely related to clique-width. We believe that this decomposition is more suitable to describe our dynamic programming.

Definition 11

Let $X \subseteq V(G)$. We say that $M \subseteq X$ is a twin-set of X if for any $v \in V(G) \setminus X$, either $M \subseteq N(v)$ or $M \cap N(v) = \emptyset$ holds. A twin-set M is called a twin-class of X if it is maximal subject to being a twin-set of X. X can be partitioned into twin-classes of X.

Definition 12

Let w be an integer. We say that $X \subseteq V(G)$ is a w-module of G if X can be partitioned into w twin-classes $\{X_1, X_2, \ldots , X_{w}\}$. A decomposition tree of G is a pair of a rooted binary tree T and a bijection $\phi$ from the set of leaves of T to V(G). For each node v of T, we denote by $L_v$ the set of leaves, each of which is either v or a descendant of v. The width of a decomposition tree $(T, \phi )$ of G is the minimum w such that for every node v in T, the set $\bigcup _{l \in L_v}\phi (l)$ is a $w_v$-module of G with $w_v \le w$. The module-width of G is the minimum t such that there is a decomposition tree of G of width w.

Rao [52] proved that clique-width and module-width are linearly related to each other. Let cw and mw be the clique-width and the module-width of G, respectively. We note that a similar terminology “modular-width” has been used in the literature, but module-width used in this paper is different from it.

Theorem 22

([52]) For every graph G, $mw \le cw \le 2mw$.

Moreover, given a w-expression tree of G, we can in time $O(n^2)$ compute a decomposition tree $(T, \phi )$ of G of width at most w and $w_v \le w$ twin-classes of $\bigcup _{l \in L_v}\phi (l)$ for each node v in T [8].

Fix a decomposition tree (T, f) of G whose width is w. Our dynamic programming algorithm runs over the nodes of the decomposition tree in a bottom-up manner. For each node v in T, we let $\{X^v_1, X^v_2, \ldots , X^v_{w_v}\}$ be the twin-classes of $\bigcup _{l \in L_v}\phi (l)$. From now on, we abuse the notation to denote $\bigcup _{l \in L_v}\phi (l)$ simply by $L_v$. A tuple of $4w_v$ integers $t= (p_1, {\overline{p}}_1, p_2, {\overline{p}}_2, \ldots , p_{w_v}, {\overline{p}}_{w_v}, c_1, {\overline{c}}_1, c_2, {\overline{c}}_2, \ldots , c_{w_v}, {\overline{c}}_{w_v})$ is valid for v if it holds that $0 \le p_i, {\overline{p}}_i \le |X^v_i|$ with $p_i + {\overline{p}}_i = |X^v_i|$ and $c_i, {\overline{c}}_i \in \{0, 1\}$ for each $1 \le i \le w_v$. For a valid tuple t for v, we say that a cut $(S, L_v \setminus S)$ of $G[L_v]$ is t-legitimate if for each $1 \le i \le w_v$, it satisfies the following conditions:

• $p_i = |S \cap X^v_i|$,

• ${\overline{p}}_i = |(L_v \setminus S) \cap X^v_i|$,

• $G[S \cap X^v_i]$ is connected if $c_i = 1$, and

• $G[(L_v \setminus S) \cap X^v_i]$ is connected if ${\overline{c}}_i = 1$.

The size of a t-legitimate cut is defined accordingly. In this section, we allow each side of a cut to be empty and the empty graph is considered to be connected. Our algorithm computes the value ${{\mathrm{mc}}}(v, t)$ that is the maximum size of a t-legitimate cut for each valid tuple t and each node v in the decomposition tree.

Leaves (Base step): For each valid tuple t for a leaf v, ${{\mathrm{mc}}}(v, t) = 0$. Note that there is only one twin-class $X^v_1 = \{v\}$ for v in this case.

Internal nodes (Induction step): Let v be an internal node of T and let a and b be the children of v in T. Consider twin-classes ${{\mathcal {X}}}^v = \{X^v_1, X^v_2, \ldots , X^v_{w_v}\}$, ${{\mathcal {X}}}^a = \{X^a_1, X^a_2, \ldots , X^a_{w_a}\}$, and ${{\mathcal {X}}}^b = \{X^b_1, X^b_2, \ldots , X^b_{w_b}\}$ of $L_v$, $L_a$, and $L_b$, respectively. Note that ${{\mathcal {X}}}^a \cup {{\mathcal {X}}}^b$ is a partition of $L_v$.

Observation 1

${{\mathcal {X}}}^v$ is a partition of $L_v$ coarser than ${{\mathcal {X}}}^a \cup {{\mathcal {X}}}^b$.

To see this, consider an arbitrary twin-class $X^a_i$ of $L_a$. By the definition of twin-sets, for every $z \in V(G) \setminus L_a$, either $X^a_i \subseteq N(z)$ or $X^a_i \cap N(z) = \emptyset$ holds. Since $V(G) \setminus L_v \subseteq V(G) \setminus L_a$, $X^a_i$ is also a twin-set of $L_v$, which implies $X^a_i$ is included in some twin-class $X^v_j$ of $L_v$. This argument indeed holds for twin-classes of $L_b$. Therefore, we have the above observation.

The intuition of our recurrence is as follows. By Observation 1, every twin-class of $L_v$ can be obtained by merging some twin-classes of $L_a$ and of $L_b$. This means that every $t_v$-legitimate cut of $G[L_v]$ for a valid tuple $t_v$ for v can be obtained from some $t_a$-legitimate cut and $t_b$-legitimate cut for valid tuples for a and b, respectively. Moreover, for every pair of twin-classes $X^a_i$ of $L_a$ and $X^b_j$ of $L_b$, either there are no edges between them or every vertex in $X^a_i$ is adjacent to every vertex in $X^b_j$ as $X^a_i$ is a twin-set of $L_v$. Therefore, the number of edges in the cutset of a cut $(S, L_v \setminus S)$ between $X^a_i$ and $X^b_j$ depends only on the cardinality of $X^a_i \cap S$ and $X^b_j \cap S$ rather than actual cuts $(S \cap X^a_i, (L_a \setminus S) \cap X^a_i)$ and $(S \cap X^b_i, (L_b \setminus S) \cap X^b_i)$.

Now, we formally describe this idea. Let $X^v$ be a twin-class of $L_v$. We denote by $I_a(X^v)$ (resp. $I_b(X^v)$) the set of indices i such that $X^a_i$ (resp. $X^b_i$) is included in $X^v$ and by ${{\mathcal {X}}}^a(X^v)$ (resp. ${{\mathcal {X}}}^b(X^v)$) the set $\{X^a_i : i \in I_a(X^v)\}$ (resp. $\{X^b_i : i \in I_b(X^v)\}$). For $X^a \in {{\mathcal {X}}}^a(X^v)$ and $X^b \in {{\mathcal {X}}}^a(X^v)$, we say that $X^a$ is adjacent to $X^b$ if every vertex in $X^a$ is adjacent to every vertex in $X^b$ and otherwise $X^a$ is not adjacent to $X^b$. This adjacency relation naturally defines a bipartite graph whose vertex set is ${{\mathcal {X}}}^a(X^v) \cup {{\mathcal {X}}}^b(X^v)$. We say that a subset of twin-classes of ${{\mathcal {X}}}^a(X^v) \cup {{\mathcal {X}}}^b(X^v)$ is non-trivially connected if it induces a connected bipartite graph with at least twin-classes. Let $S \subseteq X^v$. To make G[S] (and $G[X^v \setminus S]$) connected, the following observation is useful.

Observation 2

Suppose $S \subseteq X^v$ has a non-empty intersection with at least two twin-classes of ${{\mathcal {X}}}^a(X^v) \cup {{\mathcal {X}}}^b(X^v)$. Then, G[S] is connected if and only if the twin-classes having a non-empty intersection with S are non-trivially connected.

This observation immediately follows from the fact that every vertex in a twin-class is adjacent to every vertex in an adjacent twin-class and is not adjacent to every vertex in a non-adjacent twin-class.

Let $t_v = (p^v_1, {\overline{p}}^v_1, \ldots , p^v_{w_v}, {\overline{p}}^v_{w_v}, c^v_1, {\overline{c}}^v_2, \ldots , c^v_{w_v}, {\overline{c}}^v_{w_v})$ be a valid tuple for v. For notational convenience, we use $\mathbf{p}^v$ to denote $(p^v_1, {\overline{p}}^v_1, \ldots , p^v_{w_v}, {\overline{p}}^v_{w_v})$ and $\mathbf{c}^v$ to denote $(c^v_1, {\overline{c}}^v_2, \ldots , c^v_{w_v}, {\overline{c}}^v_{w_v})$ for each node v in T. For valid tuples $t_a = (\mathbf{p}^a, \mathbf{c}^a)$ for a and $t_b = (\mathbf{p}^b, \mathbf{c}^b)$ for b, we say that $t_v$ is consistent with the pair $(t_a, t_b)$ if for each $1\le i \le w_v$,

• C1 $p^v_i = \sum _{j \in I_a(X^v_i)} p^a_j + \sum _{j \in I_b(X^v_i)} p^b_j$;

• C2 ${\overline{p}}^v_i = \sum _{j \in I_a(X^v_i)} {\overline{p}}^a_j + \sum _{j \in I_b(X^v_i)} {\overline{p}}^b_j$;

• C3 if $c^v_i = 1$, either (1) $\{X^a_j : j \in I_a(X^v), p^a_j> 0\} \cup \{X^b_j : j \in I_b(X^v), p^b_j > 0\}$ is non-trivially connected or (2) exactly one of $\{p^s_j : s \in \{a, b\}, 1 \le j \le w_s\}$ is positive, say $p^s_j$, and $c^s_j = 1$;

• C4 if ${\overline{c}}^v_i = 1$, either (1) $\{X^a_j : j \in I_a(X^v), {\overline{p}}^a_j> 0\} \cup \{X^b_j : j \in I_b(X^v), {\overline{p}}^b_j > 0\}$ is non-trivially connected or (2) exactly one of $\{{\overline{p}}^s_j : s \in \{a, b\}, 1 \le j \le w_s\}$ is positive, say ${\overline{p}}^s_j$, and ${\overline{c}}^s_j = 1$.

Lemma 13

$$\begin{aligned} {{\mathrm{mc}}}(v, t_v) = \max _{t_a, t_b} \left( {{\mathrm{mc}}}(a, t_a) + {{\mathrm{mc}}}(b, t_b) + \sum _{\begin{array}{c} X^a_i \in {{\mathcal {X}}}^a, X^b_j \in {{\mathcal {X}}}^b\\ X^a_i, X^b_j: \text {adjacent} \end{array}} (p^a_i{\overline{p}}^b_j + p^b_j{\overline{p}}^a_i) \right) , \end{aligned}$$

where the maximum is taken over all consistent pairs $(t_a, t_b)$.

Proof

We first show that the left-hand side is at most the right-hand side. Suppose $(S, L_v \setminus S)$ be a $t_v$-legitimate cut of $G[L_v]$ whose size is equal to ${{\mathrm{mc}}}(v, t_v)$. Let $S_a = S \cap L_a$ and $S_b = S \cap L_b$. We claim that $(S_a, L_a \setminus S_a)$ is a $t_a$-legitimate cut of $G[L_a]$ for some valid tuple $t_a$ for a. This is obvious since we set $p^a_i = |S_a \cap X^a_i|$, ${\overline{p}}^a_i = |(L_a \setminus S_a) \cap X^a_i|$, $c^a_i = 1$ if $G[S_a \cap X^a_i]$ is connected, and $c^a_i = 1$ if $G[(L_a \setminus S_a) \cap X^a_i]$ is connected, which yields a valid tuple $t_a$ for a. We also conclude that $(S_b, L_b \setminus S_b)$ is a $t_b$-legitimate cut of $G[L_b]$ for some valid tuple $t_b$ for b. Moreover, the number of cut edges between twin-class $X^a_i$ of $L_a$ and twin-class $X^b_j$ of $L_b$ is $|S_a \cap X^a_i|\cdot |(L_b\setminus S_b) \cap X^b_j| + |S_b \cap X^b_j|\cdot |(L_b \setminus S_a)\cap X^a_i| = p^a_i{\overline{p}}^b_j + p^b_j{\overline{p}}^a_i$ if $X^a_i$ and $X^b_j$ is adjacent, zero otherwise. Therefore, the left-hand side is at most the right-hand side.

To show the converse direction, suppose $(S_a, L_a \setminus S_a)$ is a $t_a$-legitimate cut of $G[L_a]$ and $(S_b, L_b \setminus S_b)$ is a $t_b$-legitimate cut of $G[L_b]$, where $t_v$ is consistent with $(t_a, t_b)$ and the sizes of the cuts are ${{\mathrm{mc}}}(a, t_a)$ and ${{\mathrm{mc}}}(b, t_b)$, respectively. We claim that $(S_a \cup S_b, L_v \setminus (S_a \cup S_b))$ is a $t_v$-legitimate cut of $G[L_v]$. Since $t_v$ is consistent with $(t_a, t_b)$, for each $1 \le i \le w_v$, we have $p^v_i = \sum _{j \in I_a(X^v_i)} p^a_j + \sum _{j \in I_b(X^v_i)} p^b_j = \sum _{1 \le j \le w_a}|S_a \cap X^i_v| + \sum _{1 \le j \le w_b}|S_b \cap X^i_v| = |(S_a \cup S_b) \cap X^i_v|$. Symmetrically, we have ${\overline{p}}^i = |(L_v \setminus (S_a \cup S_b)) \cap X^v_i|$. If $c^v_i = 1$, by condition C3 of the consistency, either (1) $\{X^a_j : j \in I_a(X^v), p^a_j> 0\} \cup \{X^b_j : j \in I_b(X^v), p^b_j > 0\}$ is non-trivially connected or (2) exactly one of $\{p^s_j : s \in \{a, b\}, 1 \le j \le w_s\}$ is positive, say $p^s_j$, and $c^s_j = 1$. If (1) holds, by Observation 2, $G[(S_a \cap S_b) \cap X^i_v]$ is connected. Otherwise, as $c^s_j = 1$, $G[S_s \cap X^i_v] = G[(S_a \cup S_b) \cap X^v_i]$ is also connected. By a symmetric argument, we conclude that $G[(L_v \setminus (S_a \cup S_b)) \cap X^i_v]$ is connected if ${\overline{c}}^v_i = 1$. Therefore the cut $(S_a \cup S_b, L_v \setminus (S_a \cup S_b))$ is $t_v$-legitimate. Since the cut edges between two twin-classes of $L_a$ is counted by ${{\mathrm{mc}}}(a, t_a)$ and those between two twin-classes of $L_v$ is counted by ${{\mathrm{mc}}}(b, t_b)$. Similar to the forward direction, the number of cut edges between a twin-class of $L_a$ and a twin-class of $L_b$ can be counted by the third term in the right-hand side of the equality. Hence, the left-hand side is at least the right-hand side.$\square$

Proof of Theorem 10

From a w-expression tree of G, we can obtain a decomposition tree $(T, \phi )$ of width at most w in $O(n^2)$ time using Rao’s algorithm [52]. Based on this decomposition, we evaluate the recurrence in Lemma 13 in a bottom-up manner. The number of valid tuples for each node of T is at most $4^wn^w$. For each internal node v and for each valid tuple $t_v$ for v, we can compute ${{\mathrm{mc}}}(v, t_v)$ in $(4^wn^w)^2n^{O(1)}$ time. Overall, the running time of our algorithm is $n^{O(w)}$. Let r be the root of T. For Maximum Connected Cut, by the definition of legitimate cuts, we should take the maximum value among ${{\mathrm{mc}}}(r, (i, n-i, 1, j))$ for $1 \le i < n$ and $j \in \{0, 1\}$. Note that as $L_v$ has only one twin-class, the length of valid tuples is exactly four. For Largest Bond, we should take the maximum value among ${{\mathrm{mc}}}(r, (i, n-i, 1, 1))$ for $1 \le i < n$. $\square$