arXiv subjects

Thomas R. Cameron

Publications and source records attributed to Thomas R. Cameron.

12 recordsLinked to original sources

arXiv2025

On the minimal forts of trees

In 2018, the concept of a fort in graph theory was introduced as a non-empty subset of vertices satisfying the condition that no vertex outside the set has exactly one neighbor in the set. Since then, forts have played a significant role in characterizing zero forcing sets, modeling the zero forcing number as an integer program, and generating lower bounds for the zero forcing number of Cartesian products. Recent research has focused on the number of minimal forts, defined as those for which no proper subset is a fort. Notably, it has been established that the number of minimal forts in any graph is strictly less than Sperner's bound, a famous bound due to Emanuel Sperner (1928) on the size of a collection of subsets where no subset contains another. Moreover, lower bounds on the number of minimal forts for several families of graphs were established, and it was shown that certain families have an exponential number of minimal forts. In this article, we provide a combinatorial-cut characterization of the minimal forts in trees. Using this characterization, we derive an upper bound on the cardinality of minimal forts and a lower bound on the number of minimal forts in trees. We also characterize the trees that attain this lower bound through a four-part equivalence theorem that provides a connection to other graph parameters, such as star centers, the fort number, and the zero forcing number.

math.CO↗

arXiv2025

IP Models for Minimum Zero Forcing Sets, Forts, and Related Graph Parameters

Zero forcing is a binary coloring game on a graph where a set of filled vertices can force non-filled vertices to become filled following a color change rule. In 2008, the zero forcing number of a graph was shown to be an upper bound on its maximum nullity. In addition, the combinatorial optimization problem for the zero forcing number was shown to be NP-hard. Since then, the study of zero forcing and its related parameters has received considerable attention. In 2018, the forts of a graph were defined as non-empty subsets of vertices where no vertex outside the set has exactly one neighbor in the set. Forts have been used to model zero forcing as an integer program and provide lower bounds on the zero forcing number. To date, three integer programming models have been developed for the zero forcing number of a graph: the Infection Model, Time Step Model, and Fort Cover Model. In this article, we present variations of these models for computing the zero forcing number and related graph parameters, such as the minimum and maximum propagation times, throttling number, and fractional zero forcing number. In addition, we present several new models for computing the realized propagation time interval, all minimal forts of a graph, and the fort number of a graph. We conclude with several numerical experiments that demonstrate the effectiveness of our models when applied to small and medium order graphs. Moreover, we provide experimental evidence for several open conjectures regarding the propagation time interval, the number of minimal forts, the fort number, and the fractional zero forcing number of a graph.

math.CO↗

arXiv2025

On the forts and related parameters of the hypercube graph

In 2018, forts were defined as non-empty subsets of vertices in a graph where no vertex outside the set has exactly one neighbor in the set. Forts have since been used to characterize zero forcing sets, model zero forcing as an integer program, and provide lower bounds on the zero forcing number. In this article, we give a complete characterization of minimum forts in the hypercube graph, showing that they are automorphic to one of two sets. In contrast, non-automorphic minimum zero forcing sets are identified with distinct propagation times. We also derive the fractional zero forcing number and bounds on the fort number of the hypercube. When the hypercube's dimension is a power of two, the fort number and fractional zero forcing number are equal to the domination number, total domination number, and open packing number. Lastly, we present general constructions for minimal forts in the Cartesian product of graphs, reflecting some minimal forts of the hypercube.

math.CO↗

arXiv2024

On the number of minimal forts of a graph

In 2018, a fort of a graph was introduced as a non-empty subset of vertices in which no vertex outside of the set has exactly one neighbor in the set. Since then, forts have been used to characterize zero forcing sets, model the zero forcing number as an integer program, and generate lower bounds on the zero forcing number of a Cartesian product. In this article, we investigate the number of minimal forts of a graph, where a fort is minimal if every proper subset is not a fort. In particular, we show that the number of minimal forts of a graph of order at least six is strictly less than Sperner's bound, a famous bound due to Emanuel Sperner (1928) on the size of a collection of subsets where no subset contains another. Then, we derive an explicit formula for the number of minimal forts for several families of graphs, including the path, cycle, and spider graphs. Moreover, we show that the asymptotic growth rate of the number of minimal forts of the spider graph is bounded above by that of the path graph. We conjecture that the asymptotic growth rate of the path graph is extremal over all trees. Finally, we develop methods for constructing minimal forts of graph products using the minimal forts of the original graphs. In the process, we derive explicit formulas and lower bounds on the number of minimal forts for additional families of graphs, such as the wheel, sunlet, and windmill graphs. Most notably, we show that the family of windmill graphs has an exponential number of minimal forts with a maximum asymptotic growth rate of cube root of three, which is the largest asymptotic growth rate we have observed. We conjecture that there exist families of graphs with a larger asymptotic growth rate.

math.CO↗

arXiv2023

Forts, (fractional) zero forcing, and Cartesian products of graphs

The (disjoint) fort number and fractional zero forcing number are introduced and related to existing parameters including the (standard) zero forcing number. The fort hypergraph is introduced and hypergraph results on transversals and matchings are applied to the zero forcing number and fort number. These results are used to establish a Vizing-like lower bound for the zero forcing number of a Cartesian product of graphs for certain families of graphs, and a family of graphs achieving this lower bound is exhibited.

math.CO↗

arXiv2022

On the Laplacian spread of digraphs

In this article, we extend the notion of the Laplacian spread to simple directed graphs (digraphs) using the restricted numerical range. First, we provide Laplacian spread values for several families of digraphs. Then, we prove sharp upper bounds on the Laplacian spread for all polygonal and balanced digraphs. In particular, we show that the validity of the Laplacian spread bound for balanced digraphs is equivalent to the Laplacian spread conjecture for simple undirected graphs, which was conjectured in 2011 and proven in 2021. Moreover, we prove an equivalent statement for weighted balanced digraphs with weights between $0$ and $1$. Finally, we state several open conjectures that are motivated by empirical data.

math.CO↗

arXiv2022

Constructions of cospectral graphs with different zero forcing numbers

Several researchers have recently explored various graph parameters that can or cannot be characterized by the spectrum of a matrix associated with a graph. In this paper we show that several NP-hard zero forcing numbers are not characterized by the spectra of several types of associated matrices with a graph. In particular, we consider standard zero forcing, positive semidefinite zero forcing, and skew zero forcing, and provide constructions of infinite families of pairs of cospectral graphs which have different values for these numbers. We explore several methods for obtaining these cospectral graphs including using graph products, graph joins, and graph switching. Among these, we provide a construction involving regular adjacency cospectral graphs; the regularity of this construction also implies cospectrality with respect to several other matrices including the Laplacian, signless Laplacian, and normalized Laplacian. We also provide a construction where pairs of cospectral graphs can have an arbitrarily large difference between their zero forcing numbers.

math.CO↗

arXiv2021

On digraphs with polygonal restricted numerical range

In 2020, Cameron et al. introduced the restricted numerical range of a digraph (directed graph) as a tool for characterizing digraphs and studying their algebraic connectivity. In particular, digraphs with a restricted numerical range of a single point, a horizontal line segment, and a vertical line segment were characterized as $k$-imploding stars, directed joins of bidirectional digraphs, and regular tournaments, respectively. In this article, we extend these results by investigating digraphs whose restricted numerical range is a convex polygon in the complex plane. We provide computational methods for identifying these polygonal digraphs and show that these digraphs can be broken into three disjoint classes: normal, restricted-normal, and pseudo-normal digraphs, all of which are closed under the digraph complement. We prove sufficient conditions for normal digraphs and show that the directed join of two normal digraphs results in a restricted-normal digraph. Also, we prove that directed joins are the only restricted-normal digraphs when the order is square-free or twice a square-free number. Finally, we provide methods to construct restricted-normal digraphs that are not directed joins for all orders that are neither square-free nor twice a square-free number.

math.CO↗

arXiv2021

On the Linear Ordering Problem and the Rankability of Data

In 2019, Anderson et al. proposed the concept of rankability, which refers to a dataset's inherent ability to be meaningfully ranked. In this article, we give an expository review of the linear ordering problem (LOP) and then use it to analyze the rankability of data. Specifically, the degree of linearity is used to quantify what percentage of the data aligns with an optimal ranking. In a sports context, this is analogous to the number of games that a ranking can correctly predict in hindsight. In fact, under the appropriate objective function, we show that the optimal rankings computed via the LOP maximize the hindsight accuracy of a ranking. Moreover, we develop a binary program to compute the maximal Kendall tau ranking distance between two optimal rankings, which can be used to measure the diversity among optimal rankings without having to enumerate all optima. Finally, we provide several examples from the world of sports and college rankings to illustrate these concepts and demonstrate our results.

math.OC↗

arXiv2020

Diameter Polytopes of Feasible Binary Programs

Feasible binary programs often have multiple optimal solutions, which is of interest in applications as they allow the user to choose between alternative optima without deteriorating the objective function. In this article, we present the optimal diameter of a feasible binary program as a metric for measuring the diversity among all optimal solutions. In addition, we present the diameter binary program whose optima contains two optimal solutions of the given feasible binary program that are as diverse as possible with respect to the optimal diameter. Our primary interest is in the study of the diameter polytope, i.e., the polytope underlying the diameter binary program. Under suitable conditions, we show that much of the structure of the diameter polytope is inherited from the polytope underlying the given binary program. Finally, we apply our results on the diameter binary program and diameter polytope to cases where the given binary program corresponds to the linear ordering problem and the symmetric traveling salesman problem.

math.OC↗

arXiv2019

On the Graph Laplacian and the Rankability of Data

Recently, Anderson et al. (2019) proposed the concept of rankability, which refers to a dataset's inherent ability to produce a meaningful ranking of its items. In the same paper, they proposed a rankability measure that is based on a integer program for computing the minimum number of edge changes made to a directed graph in order to obtain a complete dominance graph, i.e., an acyclic tournament graph. In this article, we prove a spectral-degree characterization of complete dominance graphs and apply this characterization to produce a new measure of rankability that is cost-effective and more widely applicable. We support the details of our algorithm with several results regarding the conditioning of the Laplacian spectrum of complete dominance graphs and the Hausdorff distance between their Laplacian spectrum and that of an arbitrary directed graph with weights between zero and one. Finally, we analyze the rankability of datasets from the world of chess and college football.

math.CO↗

arXiv2017

On the application of Laguerre's method to the polynomial eigenvalue problem

The polynomial eigenvalue problem arises in many applications and has received a great deal of attention over the last decade. The use of root-finding methods to solve the polynomial eigenvalue problem dates back to the work of Kublanovskaya (1969, 1970) and has received a resurgence due to the work of Bini and Noferini (2013). In this paper, we present a method which uses Laguerre iteration for computing the eigenvalues of a matrix polynomial. An effective method based on the numerical range is presented for computing initial estimates to the eigenvalues of a matrix polynomial. A detailed explanation of the stopping criteria is given, and it is shown that under suitable conditions we can guarantee the backward stability of the eigenvalues computed by our method. Then, robust methods are provided for computing both the right and left eigenvectors and the condition number of each eigenpair. Applications for Hessenberg and tridiagonal matrix polynomials are given and we show that both structures benefit from substantial computational savings. Finally, we present several numerical experiments to verify the accuracy of our method and its competitiveness for solving the roots of a polynomial and the tridiagonal eigenvalue problem.

math.NA↗