New Perspectives On The Unimodality Of Domination Polynomials

Abstract

The domination polynomial of a graph $G$ is given by $D(G,x)=\sum_{k=0}^{n} d_k(G)x^k$ where $d_k(G)$ records the number of $k$-element dominating sets in $G$. A conjecture of Alikhani and Peng asserts that these polynomials have unimodal coefficient sequences. We develop three complementary perspectives that strengthen existing tools for resolving the conjecture. First, we view dominating sets as transversals of the closed neighborhood hypergraph. Motivated by the relationship between the unimodality of a polynomial and its roots, we use this perspective to expand on known root phenomena for domination polynomials. In particular, we obtain a bound on the modulus of domination roots that is linear in the maximum degree of a graph, improving related exponential bounds of Bencs, Csikvári and Regts. The hypergraph viewpoint also yields explicit combinatorial formulas for top coefficients of $D(G,x)$, extending formulas in the literature and offering fruitful ground for combinatorial approaches to the unimodality conjecture. Second, we strengthen the coefficient-ratio method of Beaton and Brown. This includes tightening their inequalities, and combining a union bound for non-dominating $k$-element sets with an overlap correction based on spanning trees. This produces a new parameter $τ_k(G)$ measuring maximal pairwise neighborhood overlap and yields an overlap-corrected sufficient criterion for unimodality. Third, we prove that the domination polynomial of threshold graphs are log-concave, and hence unimodal, by a planar network argument from total positivity. This offers a new tactic for resolving the unimodality of hereditary graph classes.