On Minimum Maximal Distance-k Matchings

TL;DR

It is proved that the recognition of k-equimatchable graphs is co-NP-complete for any fixed k ≥ 2 and the problem of finding a minimum weight maximal distance-k matching in chordal graphs is hard to approximate within a factor of e ln | V ( G ) | for a fixed e unless P = N P .

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 $δ\ln |V(G)|$ unless $\mathrm{P} = \mathrm{NP}$, where $δ$ 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.

Figures (4)

• Figure 1: Gadgets produced by SAT-to-Graph$_k$: $k = 4$ (left) and $k = 5$, assuming $x_1 \in c_1, x_1 \in c_j, \overline{x}_1 \in c_m$. Dashed lines enclose cliques.
• Figure 2: The construction used in the reduction of Set Cover to 4-WMMM
• Figure 3: The construction used in the reduction from Set Cover to 3-WIDS
• Figure 4: The construction of the graph $T(G)$

Theorems & Definitions (36)

• Claim 1
• proof
• Claim 2
• proof
• Claim 3
• proof
• Claim 4
• proof
• Theorem 1
• proof