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 \(\varDelta =4.\)
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Jakkepalli, P.K., Palagiri, V.S.R. Algorithmic aspects of 2-secure domination in graphs. J Comb Optim 42, 56–70 (2021). https://doi.org/10.1007/s10878-021-00739-9
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DOI: https://doi.org/10.1007/s10878-021-00739-9