## ALGORITHMIC COMPLEXITY OF ISOLATE SECURE DOMINATION IN GRAPHS

## ALGORITHMIC COMPLEXITY OF ISOLATE SECURE DOMINATION IN GRAPHS

*J. P. Kumar, P. V. S. Reddy*

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## Abstract

A dominating set S is an Isolate Dominating Set (IDS) if the induced sub- graph 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 γ0s(G). We give isolate secure domination number of path and cycle graphs. 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.

## Keywords

Domination, NP-complete, Secure domination, CMSOL, Isolate domination