[Paper Review] Efficient Domination for Some Subclasses of $P_6$-Free Graphs in Polynomial Time

Andreas Brandstädt, Elaine M. Eschen+1 more|arXiv (Cornell University)|Feb 28, 2015

Advanced Graph Theory Research11 references4 citations

TL;DR

This paper establishes polynomial-time algorithms for the Efficient Domination (ED) and Weighted Efficient Domination (WED) problems in specific subclasses of $P_6$-free graphs by proving that the square of such graphs is chordal or perfect under certain conditions. The key contribution is showing that ED/WED is solvable in polynomial time for $(P_6, ext{house})$-free and $(P_6, ext{bull})$-free graphs, and for $(P_6, ext{house, hole, domino})$-free graphs, resolving open cases in the complexity landscape of $P_6$-free graphs.

ABSTRACT

Let $G$ be a finite undirected graph. A vertex {\em dominates} itself and all its neighbors in $G$. A vertex set $D$ is an {\em efficient dominating set} (\emph{e.d.}\ for short) of $G$ if every vertex of $G$ is dominated by exactly one vertex of $D$. The \emph{Efficient Domination} (ED) problem, which asks for the existence of an e.d.\ in $G$, is known to be \NP-complete even for very restricted graph classes such as $P_7$-free chordal graphs. The ED problem on a graph $G$ can be reduced to the Maximum Weight Independent Set (MWIS) problem on the square of $G$. The complexity of the ED problem is an open question for $P_6$-free graphs and was open even for the subclass of $P_6$-free chordal graphs. In this paper, we show that squares of $P_6$-free chordal graphs that have an e.d. are chordal; this even holds for the larger class of ($P_6$, house, hole, domino)-free graphs. This implies that ED/WeightedED is solvable in polynomial time for ($P_6$, house, hole, domino)-free graphs; in particular, for $P_6$-free chordal graphs. Moreover, based on our result that squares of $P_6$-free graphs that have an e.d. are hole-free and some properties concerning odd antiholes, we show that squares of ($P_6$, house)-free graphs (($P_6$, bull)-free graphs, respectively) that have an e.d. are perfect. This implies that ED/WeightedED is solvable in polynomial time for ($P_6$, house)-free graphs and for ($P_6$, bull)-free graphs (the time bound for ($P_6$, house, hole, domino)-free graphs is better than that for ($P_6$, house)-free graphs). The complexity of the ED problem for $P_6$-free graphs remains an open question.

Motivation & Objective

• To resolve the open complexity status of the Efficient Domination (ED) problem for $P_6$-free graphs and related subclasses.
• To determine under which conditions the square of a $P_6$-free graph with an efficient dominating set is chordal or perfect.
• To extend polynomial-time solvability of ED/WED beyond known graph classes by leveraging structural properties of graph squares.
• To provide a dichotomy result for $P_k$-free chordal graphs by showing NP-completeness for $P_7$-free chordal graphs and polynomial-time solvability for $P_6$-free chordal graphs.
• To investigate the structure of $C_4$ realizations and odd antiholes in the square of $P_6$-free graphs to enable algorithmic reductions.

Proposed method

• Reducing the ED problem on a graph $G$ to the Maximum Weight Independent Set (MWIS) problem on $G^2$, the square of $G$.
• Proving that for $(P_6, ext{HHD})$-free graphs with an e.d., $G^2$ is chordal, enabling polynomial-time MWIS via known algorithms.
• Establishing that $G^2$ is hole-free for $P_6$-free graphs with an e.d., a key step toward perfectness.
• Analyzing $C_4$ realizations in $G^2$ to show that odd anti-holes in $G^2$ must have a very restricted structure if $G$ is $P_6$-free and has an e.d.
• Applying the Strong Perfect Graph Theorem to conclude that $G^2$ is perfect when $G$ is $(P_6, ext{house})$-free or $(P_6, ext{bull})$-free and has an e.d.
• Leveraging the fact that MWIS is polynomial-time solvable on perfect graphs to derive polynomial-time algorithms for ED/WED on the respective graph classes.

Experimental results

Research questions

• RQ1 Is the Efficient Domination problem solvable in polynomial time for $P_6$-free chordal graphs?
• RQ2 Under what conditions is the square of a $P_6$-free graph with an e.d. chordal or perfect?
• RQ3 What structural constraints do $C_4$ realizations in $G^2$ impose on $P_6$-free graphs with an e.d.?
• RQ4 Can the ED problem be solved in polynomial time for $(P_6, ext{house})$-free and $(P_6, ext{bull})$-free graphs?
• RQ5 Does the conjecture that $G^2$ is perfect for any $P_6$-free graph with an e.d. hold true?

Key findings

• For $(P_6, ext{house, hole, domino})$-free graphs, $G^2$ is chordal if $G$ has an e.d., enabling polynomial-time ED/WED via MWIS on chordal graphs.
• For $P_6$-free graphs with an e.d., $G^2$ is hole-free, a necessary but not sufficient condition for perfectness.
• For $(P_6, ext{house})$-free graphs with an e.d., $G^2$ is both hole-free and odd-antihole-free, hence perfect by the Strong Perfect Graph Theorem.
• For $(P_6, ext{bull})$-free graphs with an e.d., $G^2$ is odd-antihole-free, implying $G^2$ is perfect.
• The ED and WED problems are solvable in polynomial time for $(P_6, ext{house})$-free and $(P_6, ext{bull})$-free graphs due to the perfectness of $G^2$.
• The complexity of ED for general $P_6$-free graphs remains open, though the paper provides strong structural evidence supporting the conjecture that $G^2$ is perfect for such graphs.