On the independent domination polynomial of a graph

TL;DR

This paper studies the independent domination polynomial of graphs, exploring its roots and properties, especially in generalized compound graphs, and constructs graphs with real roots and analyzes their independent dominating sets.

Contribution

It introduces the independent domination polynomial, investigates its roots in generalized compound graphs, and constructs graphs with real roots, expanding understanding of graph domination properties.

Findings

Constructed graphs with real independence domination roots

Analyzed the number of independent dominating sets in specific graphs

Studied properties of the independent domination polynomial

Abstract

An independent dominating set of the simple graph $G=(V,E)$ is a vertex subset that is both dominating and independent in $G$. The independent domination polynomial of a graph $G$ is the polynomial $D_i(G,x)=\sum_{A} x^{|A|}$, summed over all independent dominating subsets $A\subseteq V$. A root of $D_i(G,x)$ is called an independence domination root. We investigate the independent domination polynomials of some generalized compound graphs. As consequences, we construct graphs whose independence domination roots are real. Also, we consider some certain graphs and study the number of their independent dominating sets.