On Minimum Maximal Distance-k Matchings

TL;DR

This paper investigates the complexity of finding minimal maximal distance-k matchings in graphs, introduces the class of k-equimatchable graphs, and proves several NP-hardness and inapproximability results for related problems.

Contribution

It introduces the class of k-equimatchable graphs, characterizes strongly chordal graphs with equal k-packing and k-domination numbers, and establishes new hardness and inapproximability results for various graph problems.

Findings

Recognition of k-equimatchable graphs is co-NP-complete for k ≥ 2.

Certain problems cannot be approximated within a factor of δ ln |V(G)| in polynomial time unless P=NP.

NP-hardness of minimum maximal induced matching and independent dominating set in large-girth planar graphs.

Abstract

We study the computational complexity of several problems connected with finding a maximal distance-$k$ matching of minimum cardinality or minimum weight in a given graph. We introduce the class of $k$-equimatchable graphs which is an edge analogue of $k$-equipackable graphs. We prove that the recognition of $k$-equimatchable graphs is co-NP-complete for any fixed $k \ge 2$. We provide a simple characterization for the class of strongly chordal graphs with equal $k$-packing and $k$-domination numbers. We also prove that for any fixed integer $\ell \ge 1$ the problem of finding a minimum weight maximal distance-$2\ell$ matching and the problem of finding a minimum weight $(2 \ell - 1)$-independent dominating set cannot be approximated in polynomial time in chordal graphs within a factor of $\delta \ln |V(G)|$ unless $\mathrm{P} = \mathrm{NP}$, where $\delta$ is a fixed constant (thereby improving the NP-hardness result of Chang for the independent domination case). Finally, we show the NP-hardness of the minimum maximal induced matching and independent dominating set problems in large-girth planar graphs. Note: This version (as compared to the journal submission) contains corrections to Section 4.