Dominating sets and Domination polynomials of Cycles

TL;DR

This paper investigates the structure of dominating sets in cycle graphs, derives recursive formulas for their counts, and explores the properties of the associated domination polynomials.

Contribution

It introduces a recursive method to count dominating sets in cycles and analyzes the properties of their domination polynomials.

Findings

Derived recursive formulas for dominating sets in cycles

Constructed the domination polynomial for cycles

Analyzed properties of the domination polynomial

Abstract

Let G=(V,E) be a simple graph. A set S\subset V is a dominating set of G, if every vertex in V\S is adjacent to at least one vertex in S. Let {\mathcal C}_n^i be the family of dominating sets of a cycle C_n with cardinality i, and let d(C_n,i) = |{\mathcal C}_n^i. In this paper, we construct {\mathcal C}_n^i, and obtain a recursive formula for d(C_n, i). Using this recursive formula, we consider the polynomial D(C_n, x) = \sum_{i=1}^n d(C_n, i)x^i, which we call domination polynomial of cycles and obtain some properties of this polynomial.