2 research outputs found

    On Efficient Domination for Some Classes of HH-Free Bipartite Graphs

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    A vertex set DD in a finite undirected graph GG is an {\em efficient dominating set} (\emph{e.d.s.}\ for short) of GG if every vertex of GG is dominated by exactly one vertex of DD. The \emph{Efficient Domination} (ED) problem, which asks for the existence of an e.d.s.\ in GG, is known to be \NP-complete even for very restricted HH-free graph classes such as for 2P32P_3-free chordal graphs while it is solvable in polynomial time for P6P_6-free graphs. Here we focus on HH-free bipartite graphs: We show that (weighted) ED can be solved in polynomial time for HH-free bipartite graphs when HH is P7P_7 or â„“P4\ell P_4 for fixed â„“\ell, and similarly for P9P_9-free bipartite graphs with vertex degree at most 3, and when HH is S2,2,4S_{2,2,4}. Moreover, we show that ED is \NP-complete for bipartite graphs with diameter at most 6.Comment: arXiv admin note: text overlap with arXiv:1701.0341

    Finding Efficient Domination for S1,3,3S_{1,3,3}-Free Bipartite Graphs in Polynomial Time

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    A vertex set DD in a finite undirected graph GG is an {\em efficient dominating set} (\emph{e.d.s.}\ for short) of GG if every vertex of GG is dominated by exactly one vertex of DD. The \emph{Efficient Domination} (ED) problem, which asks for the existence of an e.d.s.\ in GG, is \NP-complete for various HH-free bipartite graphs, e.g., Lu and Tang showed that ED is \NP-complete for chordal bipartite graphs and for planar bipartite graphs; actually, ED is \NP-complete even for planar bipartite graphs with vertex degree at most 3 and girth at least gg for every fixed gg. Thus, ED is \NP-complete for K1,4K_{1,4}-free bipartite graphs and for C4C_4-free bipartite graphs. In this paper, we show that ED can be solved in polynomial time for S1,3,3S_{1,3,3}-free bipartite graphs
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