Inverse problems for the number of maximal independent sets

TL;DR

This paper investigates the inverse problem of determining the minimal number of vertices in a graph with a specified number of maximal independent sets, focusing on graphs related to concatenations of periodic binary words.

Contribution

It provides asymptotic results for the minimal vertex count in graphs with a given number of maximal independent sets, specifically for certain classes of natural numbers.

Findings

Asymptotic formulas for minimal vertices given the number of maximal independent sets.

Results apply to graphs associated with concatenations of periodic binary words.

New insights into inverse problems in graph theory.

Abstract

We study the following inverse graph-theoretic problem: how many vertices should a graph have given that it has a specified value of some parameter. We obtain asymptotic for the minimal number of vertices of the graph with the given number $n$ of maximal independent sets for a class of natural numbers that can be represented as concatenation of periodic binary words.