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• W3005337246 abstract "Let $G(V,E)$ be a simple, undirected and connected graph. A dominating set $S subseteq V(G)$ is called a $2$-textit{secure dominating set} ($2$-SDS) in $G$, if for every pair of distinct vertices $u_1,u_2 in V(G)$ there exists a pair of distinct vertices $v_1,v_2 in S$ such that $v_1 in N[u_1]$, $v_2 in N[u_2]$ and $(S setminus {v_1,v_2}) cup {u_1,u_2 }$ is a dominating set in $G$. The $2$textit{-secure domination number} denoted by $gamma_{2s}(G)$, 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 $ Delta(G)+1 $ $ - $ approximation algorithm for M2SDS, where $ Delta(G) $ is the maximum degree of the input graph $ G $ and prove that M2SDS 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. A secure dominating set of a graph textit{defends} one attack at any vertex of the graph. Finally, we show that the M2SDS is APX-complete for graphs with $Delta(G)=4.$" @default.
• W3005337246 created "2020-02-14" @default.
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• W3005337246 date "2020-02-05" @default.
• W3005337246 modified "2023-09-27" @default.
• W3005337246 title "Algorithmic Aspects of 2-Secure Domination in Graphs" @default.
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