Complexity issues of perfect secure domination in graphs

Let G = ( V , E ) be a simple, undirected and connected graph. A dominating set S is called secure if for each u ∈ \ there exists v such that \{ }) ∪{ } of . If further the vertex unique, then perfect (PSDS). The domination number γ ps minimum cardinality Given graph positive integer k (PSDOM) problem to check whether has PSDS size at most In this paper, we prove PSDOM NP-complete split graphs, star convex bipartite comb planar graphs dually chordal graphs. We propose linear time algorithm solve in caterpillar trees also show solvable bounded tree-width threshold subclass Finally, problems are not equivalent computational complexity aspects.