A dichotomy for weighted efficient dominating sets with bounded degree vertices
Introduction
Throughout this note, let be a finite nontrivial undirected graph and let and . A vertex v dominates itself and its neighbors. A vertex subset 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 -complete, and it is well known that ED remains -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 , the question is whether there is a subset such that is a partition of V. Actually, the Exact Cover problem is -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 and for , let 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 , . 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 , , 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 . On the other hand, since the ED problem is -complete for graphs of maximum degree 3 [8], [11], the k-BWED problem is -complete for every .
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 and a vertex , let denote its (open) neighborhood, and let denote its closed neighborhood. For a subset , we denote by the subgraph of G induced by U. The degree of a vertex v in G is . Let denote a chordless path with k vertices, say and edges , . Clearly, a in G is an edge of G. An induced matching in G is a set of edges (i.e., 's) with pairwise distance at least 2.
Section snippets
Solving the 2-bounded WED problem
A 2-b.e.d.s. of a vertex-weighted graph G is said to be finite if it is of finite total weight. With this terminology, the 2-BWED problem can be formally stated as follows:
In what follows, we describe an algorithmic approach for the 2-BWED problem which will show that the problem is solvable in polynomial time. First, we reduce in a sequence of steps the input graph to equivalent smaller ones. This is done in Section 2.1. Second, we show that once all the reductions are performed, the graph has
The algorithm
Now we summarize the previous reduction steps and the construction of the multigraph H.
Algorithm 2-BWED:
Input: A graph with vertex weights .
Output: A 2-bounded e.d.s. D of G of minimum finite total weight if such a set exists, ‘no’, otherwise.
- (a)
Determine the degrees of all vertices . Let and and . Set for all . Delete all edges in G between vertices of Y. (This is justified by Claim 1.) If there is an isolated vertex , then return no
Acknowledgements
The authors are grateful to the anonymous reviewers for their helpful comments. The work of M.M. is supported in part by the Slovenian Research Agency (I0-0035, research program P1-0285, and research projects N1-0032, J1-7051, and J1-9110).
References (19)
- et al.
Generalized domination and efficient domination in graphs
Discrete Math.
(1996) Perfect codes in graphs
J. Comb. Theory, Ser. B
(1973)- et al.
Polynomial-time algorithms for weighted efficient domination problems in AT-free graphs and dually chordal graphs
Inf. Process. Lett.
(2015) - et al.
Weighted independent perfect domination on cocomparability graphs
Discrete Appl. Math.
(1995) - et al.
Weighted efficient domination problem on some perfect graphs
Discrete Appl. Math.
(2002) - et al.
Efficient dominating sets in graphs
Efficient domination and efficient edge domination: a brief survey
- et al.
New polynomial cases of the weighted efficient domination problem
- et al.
Perfect domination
Australas. J. Comb.
(1991)