Unimodality and monotonic portions of certain domination polynomials

Abstract

Given a simple graph G on n vertices, a subset of vertices U ⊆ 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 nondecreasing 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 ( 1 2 − 1 2 m+1 )n coefficients of their domination polynomial are non-increasing.

Domination polynomial is unimodal for large graphs with a universal vertex

“…A possible direction was suggested in [5], where the authors proposed the following conjecture. It might also be interesting to get rid of the large size constraint n ≥ 2 13 on the graph G.…”

“…Beaton and Brown computed that all graphs with at most 10 vertices and at least one universal vertex have unimodal domination polynomial, with mode either ⌊ n 2 ⌋ or ⌊ n+1 2 ⌋. The proposal was considered by Burcroff and O'Brien in [5], where they proved that a graph G on n vertices and m universal vertices satisfy…”