Algorithmic aspects of secure connected domination in graphs

Abstract

A dominating set S is an Isolate Dominating Set (IDS ) if the induced subgraph G[S] has at least one isolated vertex. In this paper, we initiate the study of new domination parameter called, isolate secure domination. An isolate dominating set S ⊆ V is an isolate secure dominating set (ISDS ), if for each vertex u ∈ V \S, there exists a neighboring vertex v of u in S such that (S \ {v}) ∪ {u} is an IDS of G. The minimum cardinality of an ISDS of G is called as an isolate secure domination number, and is denoted by γ 0s (G). Given a graph G = (V, E) and a positive integer k, the ISDM problem is to check whether G has an isolate secure dominating set of size at most k. We prove that ISDM is NP-complete even when restricted to bipartite graphs and split graphs. We also show that ISDM can be solved in linear time for graphs of bounded tree-width.