The Domination Equivalence Classes of Paths

TL;DR

This paper characterizes the equivalence classes of paths based on their domination polynomials, extending previous results to fully classify paths in terms of domination set counts.

Contribution

It determines the domination polynomial equivalence classes for all paths, providing a complete classification that extends prior work.

Findings

All path graphs are classified into their domination polynomial equivalence classes.

The paper extends previous results to include all paths in the classification.

Provides a complete characterization of paths based on domination polynomial equivalence.

Abstract

A dominating set $S$ of a graph $G$ of order $n$ is a subset of the vertices of $G$ such that every vertex is either in $S$ or adjacent to a vertex of $S$. %The domination number $G$, denoted $\gamma (G)$, is the cardinality of the smallest dominating set of $G$. The domination polynomial is defined by $D(G,x) = \sum d(G,i)x^i$ where $d(G,i)$ is the number of dominating sets in $G$ with cardinality $i$. Two graphs $G$ and $H$ are considered $\mathcal{D}$-equivalent if $D(G,x)=D(H,x)$. The equivalence class of $G$, denoted $[G]$, is the set of all graphs $\mathcal{D}$-equivalent to $G$. Extending previous results, we determine the equivalence classes of all paths.