# Well-indumatched Trees and Graphs of Bounded Girth

## TL;DR

This paper characterizes well-indumatched trees and graphs with large girth, provides a linear time recognition algorithm for trees, and explores the existence of such graphs with specific girth properties.

## Contribution

It offers a complete characterization of well-indumatched trees, a polynomial recognition algorithm, and results on the existence of well-indumatched graphs with various girth constraints.

## Key findings

- Characterization of all well-indumatched trees.
- Linear time algorithm for recognizing well-indumatched trees.
- Existence and non-existence results for well-indumatched graphs with certain girth values.

## Abstract

A graph G is called well-indumatched if all of its maximal induced matchings have the same size. In this paper we characterize all well-indumatched trees. We provide a linear time algorithm to decide if a tree is well-indumatched or not. Then, we characterize minimal well-indumatched graphs of girth at least 9 and show subsequently that for an odd integer g greater than or equal to 9 and different from 11, there is no well-indumatched graph of girth g. On the other hand, there are infinitely many well-indumatched unicyclic graphs of girth k, where k is in {3, 5, 7} or k is an even integer greater than 2. We also show that, although the recognition of well-indumatched graphs is known to be co-NP-complete in general, one can recognize in polynomial time well-indumatched graphs where the size of maximal induced matchings is fixed.