On the Parameterized Complexity of the Acyclic Matching Problem

TL;DR

This paper explores the computational complexity of finding acyclic matchings in graphs, proving NP-completeness in various graph classes and analyzing the problem's parameterized complexity, including fixed parameter tractability results.

Contribution

It establishes NP-completeness for acyclic matching in specific graph classes and provides new fixed parameter tractability results for certain parameters and graph classes.

Findings

NP-complete for planar bipartite graphs with maximum degree three and large girth

NP-complete for planar line graphs with maximum degree four

Fixed parameter tractability with respect to treewidth and certain cycle parameters

Abstract

A matching is a set of edges in a graph with no common endpoint. A matching M is called acyclic if the induced subgraph on the endpoints of the edges in M is acyclic. Given a graph G and an integer k, Acyclic Matching Problem seeks for an acyclic matching of size k in G. The problem is known to be NP-complete. In this paper, we investigate the complexity of the problem in different aspects. First, we prove that the problem remains NP-complete for the class of planar bipartite graphs of maximum degree three and arbitrarily large girth. Also, the problem remains NP-complete for the class of planar line graphs with maximum degree four. Moreover, we study the parameterized complexity of the problem. In particular, we prove that the problem is W[1]-hard on bipartite graphs with respect to the parameter k. On the other hand, the problem is fixed parameter tractable with respect to the parameters tw and (k, c4), where tw and c4 are the treewidth and the number of cycles with length 4 of the input graph. We also prove that the problem is fixed parameter tractable with respect to the parameter k for the line graphs and every proper minor-closed class of graphs (including planar graphs).