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New Perspectives On The Unimodality Of Domination Polynomials

Mohamed Omar

TL;DR

The paper tackles the unimodality problem for domination polynomials by developing three complementary approaches. First, it reframes domination as a hypergraph transversal problem using the closed-neighborhood hypergraph $\mathcal{H}_G$ and transfers zero-free regions from $I(\mathcal{H}_G,x)$ to $D(G,x)$, yielding linear-in-$\Delta$ root bounds and explicit high-coefficient formulas. Second, it strengthens Beaton–Brown’s coefficient-ratio method with an overlap correction $\tau_k(G)$, producing a refined unimodality criterion at $k=\lceil n/2\rceil$ and illustrating it on a twin-class graph family. Third, it proves that threshold graphs have log-concave, hence unimodal, domination polynomials via a total-positivity argument based on staircase decompositions, and it discusses how this method could extend to broader hereditary classes. Together, these perspectives offer new tools for proving unimodality and suggest concrete directions for future work across graph classes and domination variants.

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.

New Perspectives On The Unimodality Of Domination Polynomials

TL;DR

The paper tackles the unimodality problem for domination polynomials by developing three complementary approaches. First, it reframes domination as a hypergraph transversal problem using the closed-neighborhood hypergraph and transfers zero-free regions from to , yielding linear-in- root bounds and explicit high-coefficient formulas. Second, it strengthens Beaton–Brown’s coefficient-ratio method with an overlap correction , producing a refined unimodality criterion at and illustrating it on a twin-class graph family. Third, it proves that threshold graphs have log-concave, hence unimodal, domination polynomials via a total-positivity argument based on staircase decompositions, and it discusses how this method could extend to broader hereditary classes. Together, these perspectives offer new tools for proving unimodality and suggest concrete directions for future work across graph classes and domination variants.

Abstract

The domination polynomial of a graph is given by where records the number of -element dominating sets in . 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 , 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 -element sets with an overlap correction based on spanning trees. This produces a new parameter 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.
Paper Structure (9 sections, 16 theorems, 50 equations, 2 figures)

This paper contains 9 sections, 16 theorems, 50 equations, 2 figures.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Perspective 1: the closed neighborhood hypergraph
  4. Top coefficients from low degree structure.
  5. Roots via zero-free regions for $I(\mathcal{H},x)$.
  6. Perspective 2: overlap-corrected inequalities for unimodality
  7. Perspective 3: Total Positivity
  8. Conclusion and further directions
  9. Acknowledgments

Key Result

Proposition 2

BeatonBrown2022 Let $G$ be a graph on $n$ vertices and define $r_k(G)$ as in eq:rkDef. (i) The sequence $r_0(G),r_1(G),\dots,r_n(G)$ is nondecreasing. (ii) If $k\ge n/2$ and $r_k(G)\ge (n-k)/(k+1)$ then $d_k(G)\ge d_{k+1}(G)\ge\cdots\ge d_n(G)$. In particular, if the inequality holds at $k=\lceil n/

Figures (2)

  • Figure 1: $G_{N,s}$: two cliques $C$ and $Q$ with distinguished vertices $c_0\in C$ and $q_0\in Q$, and edges $c_0q$ for every $q\in Q\setminus\{q_0\}$.
  • Figure 2: A threshold graph with threshold order $v_1,v_2,\ldots,v_{14}$, whose dominating side is $v_3,v_5,v_9,v_{11}$.

Theorems & Definitions (34)

  • Conjecture 1: AlikhaniPeng2014
  • Proposition 2
  • Example 3
  • Theorem 4
  • proof
  • Theorem 5
  • proof
  • Theorem 6
  • proof
  • Theorem 7
  • ...and 24 more