Maximal independent sets in graphs with given matching number

TL;DR

This paper determines the maximum number of maximal independent sets in various classes of graphs based on their matching number, providing exact bounds and characterizations of extremal graphs.

Contribution

It establishes the maximum counts of maximal independent sets for different graph classes with a given matching number and characterizes the extremal graphs.

Findings

Maximum number of maximal independent sets in general graphs

Maximum in connected graphs and triangle-free graphs

Characterization of extremal graphs achieving these maxima

Abstract

A maximal independent set in a graph $G$ is an independent set that cannot be extended to a larger independent set by adding any vertex from $G$. This paper investigates the problem of determining the maximum number of maximal independent sets in terms of the matching number of a graph. We establish the maximum number of maximal independent sets for general graphs, connected graphs, triangle-free graphs, and connected triangle-free graphs with a given matching number, and characterize the extremal graphs achieving these maxima.