Abstract
The k -restricted edge-connectivity of a graph G, denoted by \(\lambda _k(G)\), is defined as the minimum size of an edge set whose removal leaves exactly two connected components each containing at least k vertices. This graph invariant, which can be seen as a generalization of a minimum edge-cut, has been extensively studied from a combinatorial point of view. However, very little is known about the complexity of computing \(\lambda _k(G)\). Very recently, in the parameterized complexity community the notion of good edge separation of a graph has been defined, which happens to be essentially the same as the k-restricted edge-connectivity. Motivated by the relevance of this invariant from both combinatorial and algorithmic points of view, in this article we initiate a systematic study of its computational complexity, with special emphasis on its parameterized complexity for several choices of the parameters. We provide a number of NP-hardness and W[1]-hardness results, as well as FPT-algorithms.
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Notes
- 1.
Given a graph G with even number of vertices, the Minimum Bisection problem consists in partitioning V(G) into two equally-sized parts minimizing the number of edges with one endpoint in each part.
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Appendices
A Dealing with the Edge Weights in the Proof of Theorem 5
As in [29], we show how to convert the instance \((G^*,k,\ell )\) of Edge-Weighted p -REC that we just constructed into an equivalent instance of p -REC such that the resulting parameter remains polynomial in n. Given \((G^*,k,\ell )\), we define \((\hat{G},k,\ell )\) as the instance of p -REC, where \(\hat{G}\) is an unweighted graph obtained from \(G^*\) as follows. We replace each vertex v of \(G^*\) with a clique \(C_v\) of size \(w_1 + \ell + 1\), and for each edge \(\{u,v\}\) of \(G^*\) with weight w, we add w pairwise disjoint edges between the cliques \(C_u\) and \(C_v\). Since no cut of size at most \(\ell \) in \(\hat{G}\) can separate a clique \(C_v\) introduced for a vertex v, it follows that \((G^*,k,\ell )\) is a Yes-instance of Edge-Weighted p -REC if and only if \((\hat{G},k,\ell )\) is a Yes-instance of p -REC. Finally, it is clear that the desired cut size \(\ell \) is still polynomial in n.
B Proof of Theorem 6
Given an instance (W, T) of 3DM with \(W=R\cup B\cup Y\), \(|R|=|B|=|Y|=m\), and \(T\subseteq R\times B\times Y\) such that each element of W appears in 2 or 3 triples only, we define an n-vertex graph \(G=(V,E)\) with maximum degree 5 as follows (see Fig. 3 for an illustration).
The set of vertices of G is
where \(T_a=\{t^a_1\ldots ,t^{a}_{|T|}\}\), \(T_b=\{b=t^b_1,t^b_2\ldots ,t^b_{|T|}\}\), \(T=\{t_1\ldots ,t_{|T|}\}\) is the set of triples, and \(\mathcal {P}=\bigcup \limits _{\sigma \in W\cup T_b\cup \{a\}} P_{\sigma }\), where \(P_{\sigma }=\{(\sigma ,t) : t=1,\ldots ,n_{\sigma }\}\) with \(n_a=(3m+|T|)n_b+5m-|T|-1\), \(n_b=2m^3\), and \(n_{\sigma }=n_b\) for every \(\sigma \in W\cup T_b\).
The set of edges of G is
where \(E_{T_a}=\{\{t_i^a,t_{i+1}^a\} : 1\le i\le |T|-1\}\), \(E_{T_b}=\{\{t_i^b,t_{i+1}^b\} : 1\le i\le |T|-1\}\), \(E_{T^+}=\{\{t_i,t_i^a\},\{t_i,t_i^b\} : 1\le i\le |T|\}\), and \(E_{\sigma }=\{\{\sigma ,(\sigma ,1)\}\}\cup \{\{(\sigma ,t),(\sigma ,t+1)\} : 1\le t\le n_{\sigma }-1\}\cup \{a,t^a_1\}\) for every \(\sigma \in W\cup T_b\cup \{a\}.\)
Note that the maximum degree of G is indeed 5. Since \(n=1+3m+3|T|+n_a+(3m+|T|)n_b\), we can observe that
Next, we show that for \(k=n/2\), G is Yes-instance of the REC problem if and only if T contains a matching covering W.
One direction is easy. Suppose first that T contains a matching M covering W. Let \(S=\{a\}\cup P_a\cup T_a\cup (T \setminus M)\). It is straightforward to check that \(|S|=n/2\) and that G[S], \( G[V\setminus S]\) are both connected.
Conversely, suppose that G can be partitioned into 2 connected subgraphs G[S], \(G[V\setminus S]\) with \(|S|=n/2\). We can assume that \(a\in S\), and then it follows that \(P_a\subseteq S\). Now \(|S \setminus (P_a\cup \{a\})|=2|T|-m<2m^3=n_b\) since \(|T|\le m^3\). As \(P_{\sigma }\subseteq S\) if and only if \(\sigma \in S\cap (W\cup T_b)\), then \(S\cap (W\cup T_b)=\emptyset \) since \(|S \setminus (P_a\cup \{a\})|<n_b\) and \(|P_{\sigma }|=n_b\) for every \(\sigma \in W\cup T_b\). Hence \(S \setminus (P_a\cup \{a\})\subseteq T\cup T_a\). Let \(M=(V\setminus S)\cap T\). Then \(|M|\le m\) since \(|S \setminus (P_a\cup \{a\})|=2|T|-m\). Finally, as \(G[V\setminus S]\) is connected and \(W\cup T_b\subseteq V\setminus S\), it follows that \(|M|\ge m\). Hence \(|M|=m\) and M must be a matching covering W.
C Computing the Minimum k-edge-degree
As it has been already mentioned, a graph may not have k-restricted edge-cuts. In fact, Esfahanian and Hakimi [15] showed that each connected graph G of order \(n\ge 4\) except a star, is \(\lambda _2\)-connected and satisfies \(\lambda _2(G)\le \xi (G)\), where \(\xi (G)\) is the minimum edge-degree of G defined as
Bonsma, Ueffing and Volkmann [5] defined an extension of the minimum edge-degree of a graph G for an integer \(k\ge 2\), called the minimum k -edge-degree, as follows:
They proved that \(\lambda _k(G)\le \xi _k(G)\) for \(1\le k \le 3\) and all graphs G aside from a class of exceptions for \(k=3\) determined in [5]. Also in the same paper, the authors give a number of examples, which show that \(\lambda _k(G)\le \xi _k(G)\) is not true in general for \(k\ge 4\). In 2005, Zhang and Yuan [31] proved that, except for the class of flowers, graphs with minimum degree greater than or equal to \(k-1\) are \(\lambda _k\)-connected. Moreover, for the same class of graphs they showed that \(\lambda _k(G)\le \xi _k(G)\) (recall that a graph G with \(|V(G)| \ge 2k\) is called a flower if it contains a cut vertex u such that every component of \(G-u\) has order at most \(k-1\)).
Therefore, given the relation between the invariants \(\lambda _k\) and \(\xi _k\), we are interested in the following parameterized problem.
Similarly to what happens with the \(\mathbf {p}\)-REC problem, the \(\mathbf {p}\)-MED problem is FPT with parameters k and \(\ell \), and W[1]-hard with parameter k.
Theorem 8
The \(\mathbf {p}\)-MED problem is FPT when parameterized by k and \(\ell \).
Proof:
The proof is based on a simple application of the splitters technique. Note that given \(G,k,\ell \), if \(\xi _k(G)\le \ell \) then G contains a vertex set X of size k whose neighborhood in \(G - X\), say \(N_X\), has size at most \(\ell \). We apply Lemma 1 with \(U = V(G)\), \(a = k\), and \(b = \ell \), obtaining in time \((k+\ell )^{O(k+\ell )} \cdot n^{O(1)}\) a family \(\mathcal{F}\) of subsets of V(G) with \(|\mathcal{F}| = (k+\ell )^{O(k+\ell )} \cdot \log n\). If \(\xi _k(G)\le \ell \), then there exists \(S \in \mathcal{F}\) containing X and disjoint from \(N_X\). Therefore, it suffices to check, for every \(S \in \mathcal{F}\), whether G[S] contains a connected component X with \(|X| = k\) and \(|[X, \overline{X}] |\le \ell \). \(\square \)
Theorem 9
The \(\mathbf {p}\)-MED problem is W[1]-hard when parameterized by k.
Proof:
The reduction is inspired by the one given in [24] to prove the W[1]-hardness of the Cutting \(\ell \) Vertices problem. Given an instance (G, k) of k-Clique such that G is r-regular for some \(r \ge k\) (the k-Clique problem is easily seen to remain W[1]-hard with this assumption [24]), we define an instance of \(\mathbf {p}\)-MED as \((G,k,\ell )\), with \(\ell := k(r-k+1)\). It is then clear that G has a clique of size k if and only if it has a vertex subset X such that \(|X|=k\), G[X] is connected, and \(|[X, \overline{X}] \le k(r-k+1)\). \(\square \)
We leave as an open problem the parameterized complexity of the \(\mathbf {p}\)-MED problem with parameter \(\ell \).
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Montejano, L.P., Sau, I. (2016). On the Complexity of Computing the k-restricted Edge-connectivity of a Graph. In: Mayr, E. (eds) Graph-Theoretic Concepts in Computer Science. WG 2015. Lecture Notes in Computer Science(), vol 9224. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-662-53174-7_16
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