Power domination polynomials of graphs

TL;DR

This paper introduces the power domination polynomial of graphs, exploring its properties, roots, and relationships with other graph polynomials, and identifies classes of graphs uniquely characterized by this polynomial.

Contribution

It defines the power domination polynomial, relates it to existing polynomials, and provides structural, extremal, and decomposition results, including explicit computations for specific graph families.

Findings

Roots and coefficients of the polynomial have structural significance.

Certain graph classes are uniquely identified by their power domination polynomial.

Decomposition formulas facilitate polynomial computation for complex graphs.

Abstract

A power dominating set of a graph is a set of vertices that observes every vertex in the graph by combining classical domination with an iterative propagation process arising from electrical circuit theory. In this paper, we study the power domination polynomial of a graph $G$ of order $n$, defined as $\mathcal{P}(G;x)=\sum_{i=1}^n p(G;i) x^i$, where $p(G;i)$ is the number of power dominating sets of $G$ of size $i$. We relate the power domination polynomial to other graph polynomials, present structural and extremal results about its roots and coefficients, and identify some graph parameters it contains. We also derive decomposition formulas for the power domination polynomial, compute it explicitly for several families of graphs, and explore graphs which can be uniquely identified by their power domination polynomials.