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An $X$-TAR (token addition/removal) reconfiguration graph has as its vertices sets that satisfy some property $X$, with an edge between two sets if one is obtained from the other by adding or removing one element. This paper considers the $X$-TAR graph for $X-$ sets of vertices of a base graph $G$ where the $X$-sets of $G$ must satisfy certain conditions. Dominating sets, power dominating sets, zero forcing sets, and positive semidefinite zero forcing sets are all examples of $X$-sets. For graphs $G$ and $G'$ with no isolated vertices, it is shown that $G$ and $G'$ have isomorphic $X$-TAR reconfiguration graphs if and only if there is a relabeling of the vertices of $G'$ such that $G$ and $G'$ have exactly the same $X$-sets. The concept of an $X$-irrelevant vertex is introduced to facilitate analysis of $X$-TAR graph isomorphisms. Furthermore, results related to the connectedness of the zero forcing TAR graph are given. We present families of graphs that exceed known lower bounds for connectedness parameters.

In many variants of the game of Cops and Robbers on graphs, multiple cops play against a single robber. In 2019, Cox and Sanaei introduced a variant of the game that gives the robber a more active role than simply evading the cop. In their version, the robber tries to damage as many vertices as possible and the cop attempts to minimize this damage. While the damage variant was originally studied with one cop and one robber, it was later extended to play with multiple cops by Carlson et. al in 2021. We take a different approach by studying the damage variant with multiple robbers against one cop. Specifically, we introduce the $s$-robber damage number of a graph and obtain a variety of bounds on this parameter. Applying these bounds, we determine the $s$-robber damage number for a variety of graph families and characterize graphs with extreme $2$-robber damage number., Comment: 16 pages, 2 figures

In this paper, we study minimal (with respect to inclusion) zero forcing sets. We first investigate when a graph can have polynomially or exponentially many distinct minimal zero forcing sets. We also study the maximum size of a minimal zero forcing set $\overline{\operatorname{Z}}(G)$, and relate it to the zero forcing number $\operatorname{Z}(G)$. Surprisingly, we show that the equality $\overline{\operatorname{Z}}(G)=\operatorname{Z}(G)$ is preserved by deleting a universal vertex, but not by adding a universal vertex. We also characterize graphs with extreme values of $\overline{\operatorname{Z}}(G)$ and explore the gap between $\overline{\operatorname{Z}}(G)$ and $\operatorname{Z}(G)$., Comment: 13 pages, 3 figures

Zero forcing is a combinatorial game played on graphs that can be used to model the spread of information with repeated applications of a color change rule. In general, a zero forcing parameter is the minimum number of initial blue vertices that are needed to eventually color every vertex blue with a given color change rule. Furthermore, the throttling number minimizes the sum of the number of initial blue vertices and the time taken for all vertices to become blue. In 2013, Barioli et al. added a new rule, called hopping, to existing color change rules in order to demonstrate that the minor monotone floor of various zero forcing parameters is itself, a zero forcing parameter. In this paper, we examine the hopping color change rule independently from the other classic rules. Specifically, we study the hopping forcing number and the hopping throttling number. We investigate the ways in which these numbers are related to various graph theory parameters (such as vertex connectivity and independence number) as well as other zero forcing parameters., Comment: 33 pages, 13 figures

Zero forcing is a process on graphs in which a color change rule is used to force vertices to become blue. The amount of time taken for all vertices in the graph to become blue is the propagation time. Throttling minimizes the sum of the number of initial blue vertices and the propagation time. In this paper, we study throttling in the context of directed graphs (digraphs). We characterize all simple digraphs with throttling number at most $t$ and examine the change in the throttling number after flipping arcs and deleting vertices. We also introduce the orientation throttling interval (OTI) of an undirected graph, which is the range of throttling numbers achieved by the orientations of the graph. While the OTI is shown to vary among different graph families, some general bounds are obtained. Additionally, the maximum value of the OTI of a path is conjectured to be achieved by the orientation of a path whose arcs alternate in direction. The throttling number of this orientation is exactly determined in terms of the number of vertices., Comment: 25 pages, 10 figures

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. In their paper, they study 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 natural parameter called the damage throttling number of a graph, denoted $\operatorname{th}_d(G)$, which optimizes the balance between the number of cops used and the number of vertices damaged in the graph. To this end, we formalize the definition of $k$-damage number, which extends the damage number to games played with $k$ cops. We show that damage throttling and cop throttling share many properties, yet they exhibit interesting differences. We prove that the damage throttling number is tightly bounded above by one less than the cop throttling number. Infinite families of examples and non-examples of tightness in this bound are given. We also find an infinite family of connected graphs $G$ of order $n$ for which $\operatorname{th}_d(G) = \Omega(n^{2/3})$., Comment: 23 pages, 8 figures

Warning. The reading of this paper will send you down many winding roads toward new and exciting research topics enumerating generalized parking functions. Buckle up!, Comment: 9 pages, 6 figures, 1 table

Given a positive integer $n$, the factorial base representation of $n$ is given by $n=\sum_{i=1}^ka_i\cdot i!$, where $a_k\neq 0$ and $0\leq a_i\leq i$ for all $1\leq i\leq k$. For $e\geq 1$, we define $S_{e,!}:\mathbb{Z}_{\geq0}\to\mathbb{Z}_{\geq0}$ by $S_{e,!}(0) = 0$ and $S_{e,!}(n)=\sum_{i=0}^{n}a_i^e$, for $n \neq 0$. For $\ell\geq 0$, we let $S_{e,!}^\ell(n)$ denote the $\ell$-th iteration of $S_{e,!}$, while $S_{e,!}^0(n)=n$. If $p\in\mathbb{Z}^+$ satisfies $S_{e,!}(p)=p$, then we say that $p$ is an $e$-power factoradic fixed point of $S_{e,!}$. Moreover, given $x\in \mathbb{Z}^+$, if $p$ is an $e$-power factoradic fixed point and if there exists $\ell\in \mathbb{Z}_{\geq 0}$ such that $S_{e,!}^\ell(x)=p$, then we say that $x$ is an $e$-power factoradic $p$-happy number. Note an integer $n$ is said to be an $e$-power factoradic happy number if it is an $e$-power factoradic $1$-happy number. In this paper, we prove that all positive integers are $1$-power factoradic happy and, for $2\leq e\leq 4$, we prove the existence of arbitrarily long sequences of $e$-power factoradic $p$-happy numbers. A curious result establishes that for any $e\geq 2$ the $e$-power factoradic fixed points of $S_{e,!}$ that are greater than $1$, always appear in sets of consecutive pairs. Our last contribution, provides the smallest sequences of $m$ consecutive $e$-power factoradic happy numbers for $2\leq e\leq 5$, for some values of $m$., Comment: 10 pages, 3 tables, 1 figure

Zero forcing can be described as a combinatorial game on a graph that uses a color change rule in which vertices change white vertices to blue. The throttling number of a graph minimizes the sum of the number of vertices initially colored blue and the number of time steps required to color the entire graph. Positive semidefinite (PSD) zero forcing is a commonly studied variant of standard zero forcing that alters the color change rule. This paper introduces a method for extending a graph using a PSD zero forcing process. Using this extension method, graphs with PSD throttling number at most $t$ are characterized as specific minors of the Cartesian product of complete graphs and trees. A similar characterization is obtained for the minor monotone floor of PSD zero forcing. Finally, the set of connected graphs on $n$ vertices with throttling number at least $n-k$ is characterized by forbidding a finite family of induced subgraphs. These forbidden subgraphs are constructed for standard throttling., Comment: 21 pages, 9 figures

Given a digraph $D$ with $m $ arcs, a bijection $\tau: A(D)\rightarrow \{1, 2, \ldots, m\}$ is an antimagic labeling of $D$ if no two vertices in $D$ have the same vertex-sum, where the vertex-sum of a vertex $u $ in $D$ under $\tau$ is the sum of labels of all arcs entering $u$ minus the sum of labels of all arcs leaving $u$. We say $(D, \tau)$ is an antimagic orientation of a graph $G$ if $D$ is an orientation of $G$ and $\tau$ is an antimagic labeling of $D$. Motivated by the conjecture of Hartsfield and Ringel from 1990 on antimagic labelings of graphs, Hefetz, M\"{u}tze, and Schwartz in 2010 initiated the study of antimagic orientations of graphs, and conjectured that every connected graph admits an antimagic orientation. This conjecture seems hard, and few related results are known. However, it has been verified to be true for regular graphs and biregular bipartite graphs. In this paper, we prove that every connected graph $G$ on $n\ge9$ vertices with maximum degree at least $n-5$ admits an antimagic orientation., Comment: 11 pages, 2 figures

The cop throttling number $th_c(G)$ of a graph $G$ for the game of Cops and Robbers is the minimum of $k + capt_k(G)$, where $k$ is the number of cops and $capt_k(G)$ is the minimum number of rounds needed for $k$ cops to capture the robber on $G$ over all possible games in which both players play optimally. In this paper, we construct a family of graphs having $th_c(G)= \Omega(n^{2/3})$, establish a sublinear upper bound on the cop throttling number, and show that the cop throttling number of chordal graphs is $O(\sqrt{n})$. We also introduce the product cop throttling number $th_c^{\times}(G)$ as a parameter that minimizes the person-hours used by the cops. This parameter extends the notion of speed-up that has been studied in the context of parallel processing and network decontamination. We establish bounds on the product cop throttling number in terms of the cop throttling number, characterize graphs with low product cop throttling number, and show that for a chordal graph $G$, $th_c^{\times}=1+rad(G)$., Comment: 19 pages, 3 figures

A power dominating set of a graph $G=(V,E)$ is a set $S\subset V$ that colors every vertex of $G$ according to the following rules: in the first timestep, every vertex in $N[S]$ becomes colored; in each subsequent timestep, every vertex which is the only non-colored neighbor of some colored vertex becomes colored. The power domination throttling number of $G$ is the minimum sum of the size of a power dominating set $S$ and the number of timesteps it takes $S$ to color the graph. In this paper, we determine the complexity of power domination throttling and give some tools for computing and bounding the power domination throttling number. Some of our results apply to very general variants of throttling and to other aspects of power domination., Comment: 19 pages

Zero forcing is a process on a graph that colors vertices blue by starting with some of the vertices blue and applying a color change rule. Throttling minimizes the sum of the size of the initial blue vertex set and the number of the time steps needed to color the graph. We study throttling for positive semidefinite zero forcing. We establish a tight lower bound on the positive semidefinite throttling number as a function of the order, maximum degree, and positive semidefinite zero forcing number of the graph, and determine the positive semidefinite throttling numbers of paths, cycles, and full binary trees. We characterize the graphs that have extreme positive semidefinite throttling numbers., Comment: 20 pages, 7 figures, in press, Discrete Appl. Math

Zero forcing is a process on a graph in which the goal is to force all vertices to become blue by applying a color change rule. Throttling minimizes the sum of the number of vertices that are initially blue and the number of time steps needed to color every vertex. We introduce a new universal definition of throttling for variants of zero forcing and the study of throttling for the minor monotone floor of zero forcing. We introduce the technique of using a zero forcing process to extend a given graph. For standard zero forcing and its floor, we use these extensions to characterize graphs with throttling number $\leq t$ as certain minors of Cartesian products of complete graphs and paths. We apply these characterizations to determine graphs with extreme throttling numbers., Comment: 17 pages, 4 figures

A vertex $v$ in a porous exponential dominating set assigns weight $\left(\tfrac{1}{2}\right)^{dist(v,u)}$ to vertex $u$. A porous exponential dominating set of a graph $G$ is a subset of $V(G)$ such that every vertex in $V(G)$ has been assigned a sum weight of at least 1. In this paper the porous exponential dominating number, denoted by $\gamma_e^*(G)$, for the graph $G = C_m \times C_n$ is discussed. Anderson et. al. proved that $\frac{mn}{15.875}\le \gamma_e^*(C_m \times C_n) \le \frac{mn}{13}$ and conjectured that $\frac{mn}{13}$ is also the asymptotic lower bound. We use a linear programing approach to sharpen the lower bound to $\frac{mn}{13.7619 + \epsilon(m,n)}$.

We consider the cop-throttling number of a graph $G$ for the game of Cops and Robbers, which is defined to be the minimum of $(k + \text{capt}_k(G))$, where $k$ is the number of cops and $\text{capt}_k(G)$ is the minimum number of rounds needed for $k$ cops to capture the robber on $G$ over all possible games. We provide some tools for bounding the cop-throttling number, including showing that the positive semidefinite (PSD) throttling number, a variant of zero forcing throttling, is an upper bound for the cop-throttling number. We also characterize graphs having low cop-throttling number and investigate how large the cop-throttling number can be for a given graph. We consider trees, unicyclic graphs, incidence graphs of finite projective planes (a Meyniel extremal family of graphs), a family of cop-win graphs with maximum capture time, grids, and hypercubes. All the upper bounds on the cop-throttling number we obtain for families of graphs are $ O(\sqrt n)$., Comment: 22 pages, 4 figures

For a graph $G$, we associate a family of real symmetric matrices, $\mathcal{S}(G)$, where for any $M \in \mathcal{S}(G)$, the location of the nonzero off-diagonal entries of $M$ are governed by the adjacency structure of $G$. The ordered multiplicity Inverse Eigenvalue Problem of a Graph (IEPG) is concerned with finding all attainable ordered lists of eigenvalue multiplicities for matrices in $\mathcal{S}(G)$. For connected graphs of order six, we offer significant progress on the IEPG, as well as a complete solution to the ordered multiplicity IEPG. We also show that while $K_{m,n}$ with $\min(m,n)\ge 3$ attains a particular ordered multiplicity list, it cannot do so with arbitrary spectrum.