On Efficient Domination for Some Classes of H-Free Chordal Graphs

Abstract

A vertex set D in a finite undirected graph G is an efficient dominating set (e.d.s. for short) of G if every vertex of G is dominated by exactly one vertex of D. The Efficient Domination (ED) problem asks for the existence of an e.d.s. in G, and the Weighted Efficient Domination (WED) problem asks for such an e.d.s. of minimum weight. While, based on [A. Brandstädt and V. Giakoumakis, Weighted Efficient Domination for (P5 + kP2)-Free Graphs in Polynomial Time, CoRR arXiv:1407.4593, 2014; A. Brandstädt, M. Milanič, and R. Nevries, New polynomial cases of the weighted efficient domination problem, extended abstract in: Conference Proceedings of MFCS 2013, LNCS 8087, 2013, 195–206; Full version: CoRR arXiv:1304.6255, 2013], for the complexity of WED on H-free graphs, a dichotomy was reached—see [A. Brandstädt and R. Mosca, Weighted efficient domination for P5-free and P6-free graphs, extended abstract in: Proceedings of WG 2016, P. Heggernes, ed., LNCS 9941, pp. 38–49, 2016; Journal version: SIAM J. Discrete Math. 30, 4 (2016) 2288–2303; D. Lokshtanov, M. Pilipczuk, and E.J. van Leeuwen, Independence and Efficient Domination on P6-Free Graphs, Proceedings of the ACM-SIAM Symposium on Discrete Algorithms (SODA) 2016, 1784–1803.]—it is still an open problem for many classes of H-free chordal graphs; a standard reduction from the NP-complete Exact Cover problem shows that WED is NP-complete for a very special subclass of chordal graphs (slightly generalizing split graphs—see the forbidden induced subgraphs such as 2P3 and seven other examples in the figure). The main results in this paper are polynomial-time solutions of WED for H-free chordal graphs, when H is co-P, net, S1,2,2, and S1,2,3.