Abstract
For a graph G = (V, E) with no isolated vertices, a set \(D\subseteq V\) is called a semipaired dominating set of G if (i)D is a dominating set of G, and (ii)D can be partitioned into two element subsets such that the vertices in each two element set are at distance at most two. The minimum cardinality of a semipaired dominating set of G is called the semipaired domination number of G, and is denoted by γpr2(G). The Minimum Semipaired Domination problem is to find a semipaired dominating set of G of cardinality γpr2(G). In this paper, we initiate the algorithmic study of the Minimum Semipaired Domination problem. We show that the decision version of the Minimum Semipaired Domination problem is NP-complete for bipartite graphs and chordal graphs. On the positive side, we present a linear-time algorithm to compute a minimum cardinality semipaired dominating set of interval graphs. We also propose a \(1+\ln (2{\Delta }+2)\)-approximation algorithm for the Minimum Semipaired Domination problem, where Δ denotes the maximum degree of the graph and show that the Minimum Semipaired Domination problem cannot be approximated within \((1-\epsilon ) \ln |V|\) for any 𝜖 > 0 unless P=NP.
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This article belongs to the Topical Collection: Special Issue on International Workshop on Combinatorial Algorithms (IWOCA 2019)
Guest Editors: Charles Colbourn, Roberto Grossi, Nadia Pisanti
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Henning, M.A., Pandey, A. & Tripathi, V. Complexity and Algorithms for Semipaired Domination in Graphs. Theory Comput Syst 64, 1225–1241 (2020). https://doi.org/10.1007/s00224-020-09988-3
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DOI: https://doi.org/10.1007/s00224-020-09988-3