Algorithmic aspects of 2-secure domination in graphs

Abstract

Let G(V, E) be a simple, connected and undirected graph. A dominating set $S \subseteq V$ is called a 2-secure dominating set (2-SDS) in G, if for each pair of distinct vertices $v_1,v_2 \in V$ there exists a pair of distinct vertices $u_1,u_2 \in S$ such that $u_1 \in N[v_1]$, $u_2 \in N[v_2]$ and $(S {\setminus } \{u_1,u_2\}) \cup \{v_1,v_2 \}$ is a dominating set in G. The size of a minimum 2-SDS in G is said to be 2-secure domination number denoted by $\gamma _{2s}(G)$. The 2-SDM problem is to check if an input graph G has a 2-SDS S, with $\vert S \vert \le k$, where $k \in \mathbb {Z}^+$. It is proved that for bipartite graphs 2-SDM is NP-complete. In this paper, we prove that the 2-SDM problem is NP-complete for planar graphs and doubly chordal graphs, a subclass of chordal graphs. We reinforce the existing NP-complete result for bipartite graphs, by proving 2-SDM is NP-complete for some subclasses of bipartite graphs specifically, comb convex bipartite and star convex bipartite graphs. We prove that this problem is linear time solvable for bounded tree-width graphs. We also show that the 2-SDM is W[2]-hard even for split graphs. The M2SDS problem is to find a 2-SDS of minimum size in the given graph. We give a $\varDelta +1$-approximation algorithm for M2SDS, where $\varDelta$ is the maximum degree of the given graph and prove that M2SDS cannot be approximated within $(1 - \epsilon ) \ln (\vert V \vert )$ for any $\epsilon > 0$ unless $NP \subseteq DTIME(\vert V \vert ^{ O(\log \log \vert V \vert )})$. Finally, we prove that the M2SDS is APX-complete for graphs with $\varDelta =4.$