A dichotomy for weighted efficient dominating sets with bounded degree vertices

Highlights

• An efficient dominating set (e.d.s.) in a graph G is a set of vertices the closed neighborhoods of which partition $V(G)$.

• An e.d.s. is said to be k-bounded if it only consists of vertices of degree at most k in G.

• The k-Bounded Weighted Efficient Domination (k-BWED) problem asks for a minimum-weight k-bounded e.d.s. in a given graph.

• A dichotomy for k-BWED is obtained: the problem is NP-hard for $k\ge 3$ and solvable in polynomial time for $k\le 2$.

Abstract

In a finite undirected graph G, a vertex v dominates itself and its neighbors. A vertex set D is an efficient dominating set (e.d.s. for short) of G if every vertex of G is dominated by exactly one vertex of D. The Efficient Domination (ED) problem, which asks for the existence of an e.d.s. in G, is well known to be $\mathbb{NP}$-complete for graphs of maximum degree 3. We say that an e.d.s. D of G is k-bounded (k-b.e.d.s. for short) if the degree of every vertex in D is at most k in G. The task of the k-Bounded Weighted Efficient Domination (k-BWED) problem is to determine whether a given vertex-weighted graph G admits a k-b.e.d.s., and if so, to compute one of minimum weight. It easily follows from the $\mathbb{NP}$-completeness of ED for graphs of maximum degree 3 that the k-BWED problem is $\mathbb{NP}$-complete for every $k\ge 3$, and clearly, the k-BWED problem is solvable in linear time for $k\le 1$. In this note, we show that the 2-BWED problem is solvable in time $\mathcal{O}(|V(G)|(|V(G)|+|E(G)|))$, thus obtaining a dichotomy of the complexity status of k-BWED over all $k\ge 0$.

Introduction

Throughout this note, let $G=(V,E)$ be a finite nontrivial undirected graph and let $|V|=n$ and $|E|=m$. A vertex v dominates itself and its neighbors. A vertex subset $D\subseteq V$ is an efficient dominating set (e.d.s. for short) of G if every vertex of G is dominated by exactly one vertex in D. Obviously, D is an e.d.s. of G if and only if the family of all closed neighborhoods of vertices in D forms a partition of V. Note that not every graph has an e.d.s.; the Efficient Dominating Set (ED) problem asks for the existence of an e.d.s. in a given graph G.

In [1], [2], it was shown that the ED problem is $\mathbb{NP}$-complete, and it is well known that ED remains $\mathbb{NP}$-complete for various graph classes while it is solvable in polynomial time on various other classes; for a list of such examples, see, e.g., [4], [5], [6].

The notion of efficient domination was introduced by Biggs [3] under the name perfect code; in [15], it is remarked that efficient dominating sets correspond to perfect 1-codes. Efficient dominating sets are also called independent perfect dominating sets in various papers – see, e.g., [7], [19] – and perfect dominating sets in [8].

The following is the weighted version of the ED problem:

The WED problem is motivated by various applications, including coding theory and resource allocation in parallel computer networks; see, e.g., [1], [2], [3], [7], [12], [13], [14], [16], [17], [18], [19]. The importance of the (W)ED problem for graphs is partly due also to the fact that ED for a graph G is a special case of the Exact Cover problem for hypergraphs (problem [SP2] of [9]); for a given hypergraph $H=(V,\mathcal{E})$, the question is whether there is a subset ${\mathcal{E}}^{\prime }\subseteq \mathcal{E}$ such that ${\mathcal{E}}^{\prime }$ is a partition of V. Actually, the Exact Cover problem is $\mathbb{NP}$-complete for hypergraphs all of whose hyperedges have size 3 as shown by Karp in [10] (called X3C in [9]). Clearly, ED is the Exact Cover problem for the closed neighborhood hypergraph of G.

In this note, we consider the following variant of the WED problem: For graph $G=(V,E)$ and for $v\in V$, let $d(v)$ denote the degree of v in G. For a non-negative integer k, we say that an e.d.s. D in G is k-bounded if for every vertex $v\in D$, $d(v)\le k$. For short, a k-bounded e.d.s. will also be referred to as a k-b.e.d.s. The task of the k-Bounded Weighted Efficient Domination (k-BWED) problem is to determine whether a given vertex-weighted graph G admits a k-b.e.d.s., and if so, to compute one of minimum weight.

Clearly, a graph G admits a 0-b.e.d.s. if and only if it is edgeless. It is also straightforward to see that G admits a 1-b.e.d.s. if and only if each connected component of G is either $K_1$, $K_2$, or the vertices of degree 1 in the component form an e.d.s. of the component. Therefore, the k-BWED problem is solvable in linear time for $k\in \{0,1\}$. On the other hand, since the ED problem is $\mathbb{NP}$-complete for graphs of maximum degree 3 [8], [11], the k-BWED problem is $\mathbb{NP}$-complete for every $k\ge 3$.

In the rest of this note, we will show that, in contrast to X3C, the 2-BWED problem can be solved in polynomial time, thus obtaining a dichotomy of the complexity status of the k-BWED problem for every value of k. This result was already mentioned in the extended abstract [6] (without proof) and in the corresponding arXiv version (without algorithm and time bound).

We will use the following notation. For a graph $G=(V,E)$ and a vertex $v\in V$, let $N(v)=\{u\in V:uv\in E\}$ denote its (open) neighborhood, and let $N[v]=\{v\}\cup N(v)$ denote its closed neighborhood. For a subset $U\subseteq V$, we denote by $G[U]$ the subgraph of G induced by U. The degree of a vertex v in G is $d(v):=|N(v)|$. Let $P_k$ denote a chordless path with k vertices, say $x_1,\dots ,x_k$ and edges $x_i{x}_{i+1}$, $1\le i\le k-1$. Clearly, a $P_2$ in G is an edge of G. An induced matching in G is a set of edges (i.e., $P_2$'s) with pairwise distance at least 2.