# Optimal Domination Polynomials

## TL;DR

This paper characterizes the conditions under which optimal graphs exist for domination polynomials and discusses the non-existence of least optimal graphs, with applications to network reliability.

## Contribution

It provides a complete characterization of graph parameters for optimal domination polynomial graphs and shows the non-existence of least optimal graphs in some cases.

## Key findings

- Identifies parameter ranges for optimal graphs.
- Proves non-existence of least optimal graphs in certain cases.
- Highlights applications to network reliability.

## Abstract

Let $G$ be a graph on $n$ vertices and $m$ edges and $D(G,x)$ the domination polynomial of $G$. In this paper we completely characterize the values of $n$ and $m$ for which optimal graphs exist for domination polynomials. We also show that there does not always exist least optimal graphs for the domination polynomial. Applications to network reliability are highlighted.