The zero forcing polynomial of a graph

Zero forcing is an iterative graph coloring process, where given a set of initially colored vertices, a colored vertex with a single uncolored neighbor causes that neighbor to become colored. A zero forcing set is a set of initially colored vertices which causes the entire graph to eventually become colored. In this paper, we study the counting problem associated with zero forcing. We introduce the zero forcing polynomial of a graph $G$ of order $n$ as the polynomial $\mathcal{Z}(G;x)=\sum_{i=1}^n z(G;i) x^i$, where $z(G;i)$ is the number of zero forcing sets of $G$ of size $i$. We characterize the extremal coefficients of $\mathcal{Z}(G;x)$, derive closed form expressions for the zero forcing polynomials of several families of graphs, and explore various structural properties of $\mathcal{Z}(G;x)$, including multiplicativity, unimodality, and uniqueness.Comment: 23 page

On the zero forcing number of the complement of graphs with forbidden subgraphs

Motivated in part by an observation that the zero forcing number for the complement of a tree on $n$ vertices is either $n-3$ or $n-1$ in one exceptional case, we consider the zero forcing number for the complement of more general graphs under some conditions, particularly those that do not contain complete bipartite subgraphs. We also move well beyond trees and completely study all of the possible zero forcing numbers for the complements of unicyclic graphs and cactus graphs.Comment: 17 pages, 8 figure

The damage throttling number of a graph

The cop throttling number of a graph, introduced in 2018 by Breen et al., optimizes the balance between the number of cops used and the number of rounds required to catch the robber in a game of Cops and Robbers. In 2019, Cox and Sanaei studied a variant of Cops and Robbers in which the robber tries to occupy (or damage) as many vertices as possible and the cop tries to minimize this damage. They investigated the minimum number of vertices damaged by the robber over all games played on a given graph G, called the damage number of G. We introduce the nat- ural parameter called the damage throttling number of a graph, denoted thd(G), which optimizes the balance between the number of cops used and the number of vertices damaged in the graph. We show that dam- age throttling and cop throttling share many properties, yet they exhibit interesting differences. We prove that thd(G) is tightly bounded above by one less than the cop throttling number. We discuss infinite families of graphs which attain equality for this bound, as well as graphs which have a greater gap between the damage throttling number and the cop throttling number. For most families of connected graphs G of order n that we consider in this paper, we prove that thd(G) = O({formula presented}). However, we also find an infinite family of connected graphs G of order n for which thd(G) = Ω(n2/3)