Some new algorithmic results on co-secure domination in graphs

Abstract

For a simple graph G=(V,E) without any isolated vertex, a co-secure dominating set S of G satisfies two properties, (i) S is a dominating set of G, (ii) for every vertex v∈S, there exists a vertex u∈V∖S such that uv∈E and (S∖{v})∪{u} is a dominating set of G. The minimum cardinality of a co-secure dominating set of G is called the co-secure domination number of G and is denoted by γcs(G). The Co-secure Domination problem is to find a co-secure dominating set of a graph G of cardinality γcs(G). The decision version of the problem is known to be NP-complete for bipartite, planar, and split graphs. On the other hand, it is also known that the Co-secure Domination problem is efficiently solvable for proper interval graphs and cographs.In an effort to reduce the complexity gap of the Co-secure Domination problem, in this paper, we work on various important graph classes. We show that the decision version of the problem remains NP-complete for doubly chordal graphs and for some subclasses of bipartite graphs, namely, chordal bipartite graphs, star-convex bipartite graphs, and comb-convex bipartite graphs. On the positive side, we give an efficient algorithm to compute the co-secure domination number of chain graphs, which is an important subclass of bipartite graphs. In addition, we show that the problem is linear-time solvable for bounded tree-width graphs and bounded clique-width graphs. Further, we establish that the computational complexity of this problem differs from that of the classical domination problem.