• Corpus ID: 238259890

# Unimodality and monotonic portions of certain domination polynomials

@inproceedings{Burcroff2021UnimodalityAM,
title={Unimodality and monotonic portions of certain domination polynomials},
author={Amanda Burcroff and Grace O'Brien},
year={2021}
}
• Published 2 October 2021
• Mathematics
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…
1 Citations

## Figures from this paper

Domination polynomial is unimodal for large graphs with a universal vertex
All graphs considered are undirected and simple. Let G = (V,E) be a graph on n vertices. For a vertex set S ⊂ V , we define the neighbor N(S) as the sets of vertices that are either in S or is

## References

SHOWING 1-10 OF 16 REFERENCES
DOMINATION POLYNOMIALS: A BRIEF SURVEY AND ANALYSIS
A dominating set S of a graph G of order n is a subset of the vertices of G such that every vertex is either in S or adjacent to a vertex of S. The domination number G, denoted γ(G), is the
DOMINATION PARAMETERS OF THE UNITARY CAYLEY GRAPH OF Z/nZ
The unitary Cayley graph of Z/nZ, denoted Xn, is the graph on {0, . . . , n − 1} where vertices a and b are adjacent if and only if gcd(a − b, n) = 1. We answer a question of Defant and Iyer by
SOME FAMILIES OF GRAPHS WHOSE DOMINATION POLYNOMIALS ARE UNIMODAL
• Mathematics
• 2017
Let $G$ be a simple graph of order $n$. The domination polynomial of $G$ is the polynomial $D(G, x)=\sum_{i=\gamma(G)}^{n} d(G,i) x^{i}$, where $d(G,i)$ is the number of dominating sets of $G$ of
On the Roots of Domination Polynomials
• Mathematics
Graphs Comb.
• 2014
This work provides an explicit family of graphs for which the domination roots are in the right half-plane, and determines the limiting curves for domination roots of complete bipartite graphs.
Dominating direct products of graphs
• Mathematics
Discret. Math.
• 2007
Introduction to Domination Polynomial of a Graph
• Mathematics
Ars Comb.
• 2014
A domination polynomial of a graph G is introduced and some properties of D(G, x) and its coefficients are obtained.
Associative graph products and their independence, domination and coloring numbers
• Mathematics
Discuss. Math. Graph Theory
• 1996
Associative products are defined using a scheme of Imrich & Izbicki [18]. These include the Cartesian, categorical, strong and lexicographic products, as well as others. We examine which product ⊗
Recurrence Relations and Splitting Formulas for the Domination Polynomial
• Mathematics
Electron. J. Comb.
• 2012
It is proved that the domination polynomial of a graph $G$ satisfies a wide range of reduction formulas and linear recurrence relations for D(G,x) for arbitrary graphs and for various special cases are shown.
(Total) Domination in Prisms
• Mathematics
Electron. J. Comb.
• 2017
It is shown that the bipartite condition is essential by constructing a (non-bipartite) graph $G$ such that $\gamma_t (G \square K_2 ) = 2\gamma(G) - k$.