Independent sets in chain cacti

TL;DR

This paper derives recurrence relations for independence polynomials in chain cacti, providing explicit formulas for independence numbers and counts of maximum independent sets, and characterizes extremal cases.

Contribution

It introduces new recurrence relations for specific classes of chain cacti and identifies extremal structures based on independent set counts.

Findings

Recurrence relations for independence polynomial in chain cacti

Explicit formulas for independence number and maximum independent sets

Characterization of extremal chain cacti (orto-chains and meta-chains)

Abstract

In this paper chain cacti are considered. First, for two specific classes of chain cacti (orto-chains and meta-chains of cycles with h vertices) the recurrence relation for independence polynomial is derived. That recurrence relation is then used in deriving explicit expressions for independence number and number of maximum independent sets for such chains. Also, the recurrence relation for total number of independent sets for such graphs is derived. Finaly, the proof is provided that orto-chains and meta-chains are the only extremal chain cacti with respect to total number of independent sets (orto-chains minimal and meta-chains maximal).