Complexity of Deciding the Equality of Matching Numbers

TL;DR

This paper investigates the computational complexity of determining when different types of matching numbers in graphs are equal, revealing NP-completeness and polynomial-time recognition results for specific graph classes.

Contribution

It characterizes the complexity of deciding equality between matching parameters in bipartite graphs and provides polynomial algorithms for certain graph classes.

Findings

Deciding equality of matching and disconnected matching numbers is NP-complete for bipartite graphs with diameter four.

Deciding equality of disconnected and induced matching numbers is co-NP-complete for bipartite graphs with diameter 3.

Polynomial-time algorithms are provided for recognizing graphs with equal matching parameters under certain degree constraints.

Abstract

A matching is said to be disconnected if the saturated vertices induce a disconnected subgraph and induced if the saturated vertices induce a 1-regular graph. The disconnected and induced matching numbers are defined as the maximum cardinality of such matchings, respectively, and are known to be NP-hard to compute. In this paper, we study the relationship between these two parameters and the matching number. In particular, we discuss the complexity of two decision problems; first: deciding if the matching number and disconnected matching number are equal; second: deciding if the disconnected matching number and induced matching number are equal. We show that given a bipartite graph with diameter four, deciding if the matching number and disconnected matching number are equal is NP-complete; the same holds for bipartite graphs with maximum degree three. We characterize diameter three graphs with equal matching number and disconnected matching number, which yields a polynomial time recognition algorithm. Afterwards, we show that deciding if the induced and disconnected matching numbers are equal is co-NP-complete for bipartite graphs of diameter 3. When the induced matching number is large enough compared to the maximum degree, we characterize graphs where these parameters are equal, which results in a polynomial time algorithm for bounded degree graphs.