On the Complexity of Computing the k-restricted Edge-connectivity of a Graph

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.

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).

Fig. 3.

Construction of the graph G in the proof of Theorem 6, with $\varDelta (G)=5$.

The set of vertices of G is

$$V=W\cup T\cup T_a\cup T_b\cup \mathcal {P}\cup \{a\},$$

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

$$E=E_I \cup E_{T_a} \cup E_{T_b}\cup E_{T^+}\bigcup \limits _{\sigma \in W\cup T_b\cup \{a\}}E_{\sigma },$$

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

$$n=2(n_a+1+2|T|-m).$$

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

$$\xi (G)=\min \{d_G(u)+d_G(v)-2: uv\in E(G)\}.$$

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:

$$\xi _{k}(G)=\min \{|[X,\overline{X}]|:|X|=k \text{ and } G[X] \text{ is } \text{ connected }\}.$$

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 $\ell$.