Maximal Independent Sets in Polygonal Cacti

TL;DR

This paper investigates the enumeration and asymptotic behavior of maximal independent sets in regular polygonal cacti graphs using generating functions and recurrence relations.

Contribution

It introduces recurrence relations for counting maximal independent sets in regular polygonal cacti and analyzes their asymptotic growth using meromorphic functions.

Findings

Derived recurrence relations for n-gonal cacti

Established asymptotic behaviors of maximal independent sets

Applied generating functions to combinatorial graph enumeration

Abstract

Counting the number of maximal independent sets of graphs was started over $50$ years ago by Erd\H{o}s and Mooser. The problem has been continuously studied with a number of variations. Interestingly, when the maximal condition of an independent set is removed, such the concept presents one of topological indices in molecular graphs, the so called Merrifield-Simmons index. In this paper, we applied the concept of bivariate generating function to establish the recurrence relations of the numbers of maximal independent sets of regualr $n$-gonal cacti when $3 \leq n \leq 6$. By the ideas on meromorphic functions and the growth of power series coefficients, the asymptotic behaviors through simple functions of these recurrence relations have been established.