Algorithmic Complexity of Isolate Secure Domination in Graphs

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 $S\subseteq V$ is an isolate secure dominating set (ISDS), if for each vertex $u \in V \setminus S$, there exists a neighboring vertex $v$ of $u$ in $S$ such that $(S \setminus \{v\}) \cup \{u\}$ 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 $\gamma_{0s}(G)$. 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.Comment: arXiv admin note: substantial text overlap with arXiv:2002.00002; text overlap with arXiv:2001.1125

Algorithmic Aspects of Secure Connected Domination in Graphs

Let $G = (V,E)$ be a simple, undirected and connected graph. A connected dominating set $S \subseteq V$ is a secure connected dominating set of $G$, if for each $u \in V\setminus S$, there exists $v\in S$ such that $(u,v) \in E$ and the set $(S \setminus \{ v \}) \cup \{ u \}$ is a connected dominating set of $G$. The minimum size of a secure connected dominating set of $G$ denoted by $\gamma_{sc} (G)$, is called the secure connected domination number of $G$. Given a graph $G$ and a positive integer $k,$ the Secure Connected Domination (SCDM) problem is to check whether $G$ has a secure connected dominating set of size at most $k.$ In this paper, we prove that the SCDM problem is NP-complete for doubly chordal graphs, a subclass of chordal graphs. We investigate the complexity of this problem for some subclasses of bipartite graphs namely, star convex bipartite, comb convex bipartite, chordal bipartite and chain graphs. The Minimum Secure Connected Dominating Set (MSCDS) problem is to find a secure connected dominating set of minimum size in the input graph. We propose a $(\Delta(G)+1)$ - approximation algorithm for MSCDS, where $\Delta(G)$ is the maximum degree of the input graph $G$ and prove that MSCDS cannot be approximated within $(1 -\epsilon) ln(| V |)$ for any $\epsilon > 0$ unless $NP \subseteq DTIME(| V |^{O(log log | V |)})$ even for bipartite graphs. Finally, we show that the MSCDS is APX-complete for graphs with $\Delta(G)=4$

Algorithmic complexity of isolate secure domination in graphs

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 subset of V is an isolate secure dominating set (ISDS), if for each vertex u is an element of V \ S, there exists a neighboring vertex v of u in S such that (S \ {v}) boolean OR {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 gamma(0s) (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.Publisher's Versio