On Almost Well-Covered Graphs of Girth at Least 6

TL;DR

This paper studies graphs that are nearly well-covered, specifically those with an independence gap of one and girth at least 6, providing structural insights and polynomial-time recognition algorithms for certain classes.

Contribution

It characterizes almost well-covered graphs of girth at least 6, identifies structural properties, and develops polynomial-time algorithms for recognizing specific subclasses.

Findings

Graphs of girth at least 6 have at most two vertices adjacent to exactly two leaves.

Characterizations for graphs with one or two such vertices are provided.

Polynomial-time recognition algorithms are developed for certain graph classes.

Abstract

We consider a relaxation of the concept of well-covered graphs, which are graphs with all maximal independent sets of the same size. The extent to which a graph fails to be well-covered can be measured by its independence gap, defined as the difference between the maximum and minimum sizes of a maximal independent set in $G$. While the well-covered graphs are exactly the graphs of independence gap zero, we investigate in this paper graphs of independence gap one, which we also call almost well-covered graphs. Previous works due to Finbow et al. (1994) and Barbosa et al. (2013) have implications for the structure of almost well-covered graphs of girth at least $k$ for $k\in \{7,8\}$. We focus on almost well-covered graphs of girth at least $6$. We show that every graph in this class has at most two vertices each of which is adjacent to exactly $2$ leaves. We give efficiently testable characterizations of almost well-covered graphs of girth at least $6$ having exactly one or exactly two such vertices. Building on these results, we develop a polynomial-time recognition algorithm of almost well-covered $\{C_3,C_4,C_5,C_7\}$-free graphs.