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Algorithmic Aspects of Some Variants of Domination in Graphs

A set $S \subseteq V$ is a dominating set in G if for every u \in V \ S, there exists $v \in S$ such that $(u,v) \in E$, i.e., $N[S] = V$. A dominating set $S$ is an Isolate Dominating Set} (IDS) if the induced subgraph $G[S]$ has at least one isolated vertex. It is known that Isolate Domination Decision problem (IDOM) is NP-complete for bipartite graphs. In this paper, we extend this by showing that the IDOM is NP-complete for split graphs and perfect elimination bipartite graphs, a subclass of bipartite graphs. A set $S \subseteq V$ is an independent set if G[S] has no edge. A set S \subseteq V is a secure dominating set of $G$ if, for each vertex $u \in V \setminus S$, there exists a vertex $v \in S$ such that $ (u,v) \in E $ and $(S \ \{v\}) \cup \{u\}$ is a dominating set of $G$. In addition, we initiate the study of a new domination parameter called, independent secure domination. A set $S\subseteq V$ is an Independent Secure Dominating Set (InSDS) if $S$ is an independent set and a secure dominating set of $G$. The minimum size of an InSDS in $G$ is called the independent secure domination number of $G$ and is denoted by $γ_{is}(G)$. Given a graph $ G$ and a positive integer $ k,$ the InSDM problem is to check whether $ G $ has an independent secure dominating set of size at most $ k.$ We prove that InSDM is NP-complete for bipartite graphs and linear time solvable for bounded tree-width graphs and threshold graphs, a subclass of split graphs. The MInSDS problem is to find an independent secure dominating set of minimum size, in the input graph. Finally, we prove that the MInSDS problem is APX-hard for graphs with maximum degree $5.$