On upper bounds for parameters related to construction of special maximum matchings

TL;DR

This paper establishes upper bounds relating the largest and smallest maximum matchings in graphs, characterizes graphs where these bounds are tight, and explores the computational complexity of testing these properties.

Contribution

It proves new bounds on maximum matchings, characterizes graphs with specific matching properties, and analyzes the complexity of testing these properties.

Findings

Proves that L(G) ≤ 2l(G) for any graph G.

Shows that L(G) ≤ (3/2)l(G) if G has a perfect matching.

Provides a polynomial algorithm to test when L(G)=2l(G).

Abstract

For a graph $G$ let $L(G)$ and $l(G)$ denote the size of the largest and smallest maximum matching of a graph obtained from $G$ by removing a maximum matching of $G$. We show that $L(G)\leq 2l(G),$ and $L(G)\leq (3/2)l(G)$ provided that $G$ contains a perfect matching. We also characterize the class of graphs for which $L(G)=2l(G)$. Our characterization implies the existence of a polynomial algorithm for testing the property $L(G)=2l(G)$. Finally we show that it is $NP$-complete to test whether a graph $G$ containing a perfect matching satisfies $L(G)=(3/2)l(G)$.