Maximizing the number of maximal independent sets of a fixed size

TL;DR

This paper investigates the maximum number of maximal independent sets of a fixed size in graphs of a given order, generalizing classical extremal graph results and identifying the extremal structures.

Contribution

It extends classical extremal graph theory by determining the maximum number of fixed-size maximal independent sets and characterizing the extremal graphs.

Findings

Identifies extremal graphs for fixed-size maximal independent sets

Generalizes classical maximum independent set results

Provides formulas for maximum counts in graphs of order n

Abstract

For a fixed graph G, a maximal independent set is an independent set that is not a proper subset of any other independent set. P. Erd\"os, and independently, J. W. Moon and L. Moser, and R. E. Miller and D. E. Muller, determined the maximum number of maximal independent sets in a graph on n vertices, as well as the extremal graphs. In this paper we maximize the number of maximal independent sets of a fixed size for all graphs of order n and determine the extremal graphs. Our result generalizes the classical result.