Algorithmic Aspects of 2-Secure Domination in Graphs
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Let G(V,E) be a simple, undirected and connected graph. A dominating set S ⊆ V(G) is called a 2-secure dominating set (2-SDS) in G, if for every pair of distinct vertices u_1,u_2 ∈ V(G) there exists a pair of distinct vertices v_1,v_2 ∈ S such that v_1 ∈ N[u_1], v_2 ∈ N[u_2] and (S ∖{v_1,v_2}) ∪{u_1,u_2 } is a dominating set in G. The 2-secure domination number denoted by γ_2s(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 Δ(G)+1- approximation algorithm for M2SDS, where Δ(G) is the maximum degree of the input graph G and prove that M2SDS cannot be approximated within (1 - ϵ) ln(| V | ) for any ϵ > 0 unless NP ⊆ DTIME(| V |^ O(loglog | V | )). 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 Δ(G)=4.
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