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 γ0s(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.Comment: arXiv admin note: substantial text overlap with arXiv:2002.00002;
text overlap with arXiv:2001.1125