On Minimum Maximal Distance-k Matchings

Abstract

We study the computational complexity of several problems connected withfinding a maximal distance-$k$ matching of minimum cardinality or minimumweight in a given graph. We introduce the class of $k$-equimatchable graphswhich is an edge analogue of $k$-equipackable graphs. We prove that therecognition of $k$-equimatchable graphs is co-NP-complete for any fixed $k \ge2$. We provide a simple characterization for the class of strongly chordalgraphs with equal $k$-packing and $k$-domination numbers. We also prove thatfor any fixed integer $\ell \ge 1$ the problem of finding a minimum weightmaximal distance-$2\ell$ matching and the problem of finding a minimum weight$(2 \ell - 1)$-independent dominating set cannot be approximated in polynomialtime in chordal graphs within a factor of $\delta \ln |V(G)|$ unless$\mathrm{P} = \mathrm{NP}$, where $\delta$ is a fixed constant (therebyimproving the NP-hardness result of Chang for the independent domination case).Finally, we show the NP-hardness of the minimum maximal induced matching andindependent dominating set problems in large-girth planar graphs.Comment: 15 pages, 4 figure