arXiv subjects

MacKenzie Carr

Publications and source records attributed to MacKenzie Carr.

9 recordsLinked to original sources

arXiv2026

Eternal domination in Cayley graphs

Eternal domination is a process in which a set of guards occupying a dominating set on a graph protects against an infinite sequence of attacks. After a vertex is attacked, one guard must move along an edge to the attacked vertex and each of the remaining guards may move along an edge so that the guards again occupy a dominating set on the graph and can defend the next attack. The minimum number of guards needed in a graph $\Gamma$ is the eternal domination number, denoted by $\gamma_{\mathrm{all}}^\infty(\Gamma)$. In this paper, we show that the eternal domination number of a vertex-transitive graph with an efficient dominating set is equal to its domination number. We show that a Cayley graph on a generalized dihedral group whose connection set contains few or many reflections is efficiently dominated. Then, we provide an infinite family of connected Cayley graphs for which $\gamma_{\mathrm{all}}^{\infty}(\Gamma) = \gamma(\Gamma)+1$, generalizing a result of [Braga et al., J. Combin. Math. Combin. Comput. 96 (2016), 13--22]. Finally, we build an infinite family of connected Cayley graphs with $\gamma_{\mathrm{all}}^\infty(\Gamma) \geq \gamma(\Gamma)+2$.

math.CO↗

arXiv2026

Cooling graph products

The cooling number measures the speed at which a slow-moving influence or contagion spreads on a graph. In this paper, we investigate the cooling number of four classical graph products: the Cartesian product, the strong product, the lexicographic product, and the direct product. We also determine the cooling number of a disconnected graph in terms of the cooling numbers of its components. We conclude with open problems.

math.CO↗

arXiv2026

2-cell embeddings of cubic graphs I. The unstable dual

In this paper, the first of a two-part series, we explore 2-cell embeddings of cubic graphs, particularly those with small genus. Using local rotations, we introduce a new way of describing the space of 2-cell embeddings and their mutual relationship for any fixed (cubic) graph. We introduce the unstable dual of an embedding of a cubic graph, a subgraph of the dual graph, and describe how the genus of the corresponding embedding can be recovered from properties of the unstable dual. Finally, we characterize the unstable duals of embeddings with genus at most 2 of cubic cyclically 5-edge connected planar graphs and use these to generate those of genus 3 that have connectivity at most 2.

math.CO↗

arXiv2026

The Iterated Local Model for tournaments

Transitivity is a central, generative principle in social and other complex networks, capturing the tendency for two nodes with a common neighbor to form a direct connection. We propose a new model for highly dense, complex networks based on transitivity, called the Iterated Local Model Tournament (ILMT). In ILMT, we iteratively apply transitivity to form new tournaments by cloning nodes and their adjacencies, and either preserving or reversing the orientation of existing arcs between clones. The resulting model generates tournaments with small diameters and high connectivity as observed in real-world complex networks. We analyze subtournaments or motifs in the ILMT model and their universality properties. For many parameter choices, the model generates sequences of quasirandom tournaments. We also study the graph-theoretic properties of ILMT tournaments, including their cop number, domination number, and chromatic number. We finish with a set of open problems and variants of the ILMT model for oriented graphs.

cs.SI↗

arXiv2025

Reconstruction of C_4-free graphs from the set of closed neighborhoods and digital convexity

Fomin, Kratochv\'il, Lokshtanov, Mancini, and Telle showed that every $C_{4}$-free graph is reconstructible from the \emph{multiset} of closed neighborhoods. We strengthen their result proving that every $C_{4}$-free graph is reconstructible from the \emph{set} of closed neighborhoods. This extends the work of Lafrance et al.\ by showing that all $C_{4}$-free graphs, and hence all graphs of girth at least five, are reconstructible from their digitally convex sets. A subset $S$ of vertices in a graph $G$ is digitally convex if, for every vertex $v \notin S$, there is a private neighbor of $v$. We establish that reconstruction from digitally convex sets is equivalent to reconstruction from the set of closed neighborhoods.

math.CO↗

arXiv2024

Trail Trap: a variant of Partizan Edge Geography

We study a two-player game played on undirected graphs called {\sc Trail Trap}, which is a variant of a game known as {\sc Partizan Edge Geography}. One player starts by choosing any edge and moving a token from one endpoint to the other; the other player then chooses a different edge and does the same. Alternating turns, each player moves their token along an unused edge from its current vertex to an adjacent vertex, until one player cannot move and loses. We present an algorithm to determine which player has a winning strategy when the graph is a tree and partially characterize the trees on which a given player wins. Additionally, we show that it is NP-hard to determine if Player~2 has a winning strategy on {\sc Trail Trap} from the starting position, even for connected bipartite planar graphs with maximum degree $4$. We determine which player has a winning strategy for certain subclasses of complete bipartite graphs and grid graphs, and we propose several open problems for further study.

math.CO↗

arXiv2023

On the interval coloring impropriety of graphs

An improper interval (edge) coloring of a graph $G$ is an assignment of colors to the edges of $G$ satisfying the condition that, for every vertex $v \in V(G)$, the set of colors assigned to the edges incident with $v$ forms an integral interval. An interval coloring is $k$-improper if at most $k$ edges with the same color all share a common endpoint. The minimum integer $k$ such that there exists a $k$-improper interval coloring of the graph $G$ is the interval coloring impropriety of $G$, denoted by $\mu_{int}(G)$. In this paper, we provide a construction of an interval coloring of a subclass of complete multipartite graphs. This provides additional evidence to the conjecture by Casselgren and Petrosyan that $\mu_{int}(G)\leq 2$ for all complete multipartite graphs $G$. Additionally, we determine improved upper bounds on the interval coloring impropriety of several classes of graphs, namely 2-trees, iterated triangulations, and outerplanar graphs. Finally, we investigate the interval coloring impropriety of the corona product of two graphs, $G\odot H$.

math.CO↗

arXiv2022

Genus Polynomials of Cubic Graphs with Non-Real Roots

Given a graph $G$, its genus polynomial is $\Gamma_G(x) = \sum_{k\geq 0} g_k(G)x^k$, where $g_k(G)$ is the number of 2-cell embeddings of $G$ in an orientable surface of genus $k$. The Log-Concavity Genus Distribution (LCGD) Conjecture states that the genus polynomial of every graph is log-concave. It was further conjectured by Stahl that the genus polynomial of every graph has only real roots, however this was later disproved. We identify several examples of cubic graphs whose genus polynomials, in addition to having at least one non-real root, have a quadratic factor that is non-log-concave when factored over the real numbers.

math.CO↗

arXiv2020

Enumerating the Digitally Convex Sets of Powers of Cycles and Cartesian Products of Paths and Complete Graphs

Given a finite set $V$, a convexity $\mathscr{C}$, is a collection of subsets of $V$ that contains both the empty set and the set $V$ and is closed under intersections. The elements of $\mathscr{C}$ are called convex sets. The digital convexity, originally proposed as a tool for processing digital images, is defined as follows: a subset $S\subseteq V(G)$ is digitally convex if, for every $v\in V(G)$, we have $N[v]\subseteq N[S]$ implies $v\in S$. The number of cyclic binary strings with blocks of length at least $k$ is expressed as a linear recurrence relation for $k\geq 2$. A bijection is established between these cyclic binary strings and the digitally convex sets of the $(k-1)^{th}$ power of a cycle. A closed formula for the number of digitally convex sets of the Cartesian product of two complete graphs is derived. A bijection is established between the digitally convex sets of the Cartesian product of two paths, $P_n \square P_m$, and certain types of $n \times m$ binary arrays.

math.CO↗