NDLI: Weighted Efficient Domination for $P_5$-Free and $P

Content Provider	Society for Industrial and Applied Mathematics (SIAM)
Author	Brandstädt, Andreas Mosca, Raffaele
Copyright Year	2016
Abstract	In a finite undirected graph $G=(V,E)$, a vertex $v \in V$ dominates itself and its neighbors in $G$. A vertex set $D \subseteq V$ is an efficient dominating set (e.d.s. for short) of $G$ if every $v \in V$ is dominated in $G$ by exactly one vertex of $D$. The Efficient Domination (ED) problem, which asks for the existence of an e.d.s. in $G$, is known to be NP-complete for $P_7$-free graphs and solvable in polynomial time for $P_5$-free graphs. The $P_6$-free case was the last open question for the complexity of ED on $F$-free graphs. Recently, Lokshtanov, Pilipczuk, and van Leeuwen showed that weighted ED is solvable in polynomial time for $P_6$-free graphs, based on their quasi-polynomial algorithm for the Maximum Weight Independent Set problem for $P_6$-free graphs. Independently, by a direct approach which is simpler and faster, we found an ${\cal O}(n^5 m)$ time solution for weighted ED on $P_6$-free graphs. Moreover, we show that weighted ED is solvable in linear time for $P_5$-free graphs which solves another open question for the complexity of (weighted) ED. The result for $P_5$-free graphs is based on modular decomposition.
Starting Page	2288
Ending Page	2303
Page Count	16
File Format	PDF
ISSN	08954801
DOI	10.1137/15M1039821
e-ISSN	10957146
Journal	SIAM Journal on Discrete Mathematics (SJDMEC)
Issue Number	4
Volume Number	30
Language	English
Publisher	Society for Industrial and Applied Mathematics
Publisher Date	2016-12-21
Access Restriction	Subscribed
Subject Keyword	$P_6$-free graphs $P_5$-free graphs linear time algorithm Factorization, matching, partitioning, covering and packing Dominating sets, independent sets, cliques weighted efficient domination Graph theory Graph algorithms polynomial time algorithm
Content Type	Text
Resource Type	Article
Subject	Mathematics

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