Algorithmic Aspects of 2-Secure Domination in Graphs
Abstract
Let G(V,E) be a simple, undirected and connected graph. A dominating set S ⊂eq V(G) is called a 2-secure dominating set (2-SDS) in G, if for every pair of distinct vertices u1,u2 ∈ V(G) there exists a pair of distinct vertices v1,v2 ∈ S such that v1 ∈ N[u1], v2 ∈ N[u2] and (S \v1,v2\) \u1,u2 \ is a dominating set in G. The 2-secure domination number denoted by γ2s(G), equals the minimum cardinality of a 2-SDS in G. Given a graph G and a positive integer k, the 2 -Secure Domination ( 2 -SDM) problem is to check whether G has a 2 -secure dominating set of size at most k. It is known that 2 -SDM is NP-complete for bipartite graphs. 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 strengthen the NP-complete result for bipartite graphs, by proving this problem is NP-complete for some subclasses of bipartite graphs namely, star convex bipartite, comb convex bipartite graphs. We prove that 2 -SDM 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 Minimum 2 -Secure Dominating Set (M2SDS) problem is to find a 2 -secure dominating set of minimum size in the input graph. We propose a (G)+1 - approximation algorithm for M2SDS, where (G) is the maximum degree of the input graph G and prove that M2SDS cannot be approximated within (1 - ε) (| V | ) for any ε > 0 unless NP ⊂eq DTIME(| V | O( | V | )) . % even for bipartite graphs. A secure dominating set of a graph defends one attack at any vertex of the graph. Finally, we show that the M2SDS is APX-complete for graphs with (G)=4.