Perfectly Matched Sets in Graphs: Parameterized and Exact Computation

TL;DR

This paper investigates the computational complexity of finding perfectly matched sets in graphs, providing fixed-parameter tractable algorithms for certain parameters, proving hardness results, and presenting an exact exponential algorithm.

Contribution

It introduces new FPT algorithms for PMS based on graph parameters, proves W[1]-hardness and kernelization lower bounds, and offers an exact exponential algorithm.

Findings

PMS is W[1]-hard parameterized by solution size k.

PMS is FPT with respect to clique-width, distance to cluster, co-cluster, and treewidth.

PMS remains NP-hard on planar graphs.

Abstract

In an undirected graph $G=(V,E)$, we say $(A,B)$ is a pair of perfectly matched sets if $A$ and $B$ are disjoint subsets of $V$ and every vertex in $A$ (resp. $B$) has exactly one neighbor in $B$ (resp. $A$). The size of a pair of perfectly matched sets $(A,B)$ is $|A|=|B|$. The PERFECTLY MATCHED SETS problem is to decide whether a given graph $G$ has a pair of perfectly matched sets of size $k$. We show that PMS is $W[1]$-hard when parameterized by solution size $k$ even when restricted to split graphs and bipartite graphs. We observe that PMS is FPT when parameterized by clique-width, and give FPT algorithms with respect to the parameters distance to cluster, distance to co-cluster and treewidth. Complementing FPT results, we show that PMS does not admit a polynomial kernel when parameterized by vertex cover number unless $NP\subseteq coNP/poly$. We also provide an exact exponential algorithm running in time $O^*(1.966^n)$. Finally, considering graphs with structural assumptions, we show that PMS remains $NP$-hard on planar graphs.