Unimodality and monotonic portions of certain domination polynomials

TL;DR

This paper investigates the unimodality and monotonic properties of domination polynomials across various graph families, providing new proofs and extending understanding of their coefficient behaviors.

Contribution

It establishes unimodality for several classes of graphs and identifies conditions under which polynomial coefficients are non-increasing, advancing the theoretical understanding of domination polynomials.

Findings

Proved unimodality for spider graphs with up to 400 legs

Established unimodality for lollipop graphs and certain graph products

Identified large portions of coefficients that are non-increasing in graphs with universal vertices

Abstract

Given a simple graph $G$ on $n$ vertices, a subset of vertices $U \subseteq V(G)$ is dominating if every vertex of $V(G)$ is either in $U$ or adjacent to a vertex of $U$. The domination polynomial of $G$ is the generating function whose coefficients are the number of dominating sets of a given size. We show that the domination polynomial is unimodal, i.e., the coefficients are non-decreasing and then non-increasing, for several well-known families of graphs. In particular, we prove unimodality for spider graphs with at most $400$ legs (of arbitrary length), lollipop graphs, arbitrary direct products of complete graphs, and Cartesian products of two complete graphs. We show that for every graph, a portion of the coefficients are non-increasing, where the size of the portion depends on the upper domination number, and in certain cases this is sufficient to prove unimodality. Furthermore, we study graphs with $m$ universal vertices, i.e., vertices adjacent to every other vertex, and show that the last $(\frac{1}{2} - \frac{1}{2^{m+1}}) n$ coefficients of their domination polynomial are non-increasing.