arXiv subjects

Franklin Kenter

Publications and source records attributed to Franklin Kenter.

15 recordsLinked to original sources

arXiv2025

Leaky Forcing: Extending Zero Forcing Results to a Fault-Tolerant Setting

We study a recent variation of zero forcing called leaky forcing. Zero forcing is a propagation process on a network whereby some nodes are initially blue with all others white. Blue vertices can "force" a white neighbor to become blue if all other neighbors are blue. The goal is to find the minimum number of initially blue vertices to eventually force all vertices blue after exhaustively applying the forcing rule above. Leaky forcing is a fault-tolerant variation of zero forcing where certain vertices (not necessarily initially blue) cannot force. The goal in this context is to find the minimum number of initially blue vertices needed that can eventually force all vertices to be blue, regardless of which small number of vertices can't force. This work extends results from zero forcing in terms of leaky forcing. In particular, we provide a complete determination of leaky forcing numbers for all unicyclic graphs and upper bounds for generalized Petersen graphs. We also provide bounds for the effect of both edge removal and vertex removal on the $\ell$-leaky forcing number. Finally, we completely characterize connected graphs that have the minimum and maximum possible $1$-leaky forcing number (i.e., when $Z_{1}(G) = 2$ and when $Z_{1}(G) = |V(G)|-1$).

math.CO↗

arXiv2023

Zero Forcing on 2-connected Outerplanar Graphs

We determine upper and lower bounds on the zero forcing number of 2-connected outerplanar graphs in terms of the structure of the weak dual. We show that the upper bound is always at most half the number of vertices of the graph. This work generalizes work of Hern\'andez, Ranilla and Ranilla-Cortina who proved a similar result for maximal outerplanar graphs.

math.CO↗

arXiv2023

Improved bounds on the cop number when forbidding a minor

Andreae (1986) proved that the cop number of connected $H$-minor-free graphs is bounded for every graph $H$. In particular, the cop number is at most $|E(H-h)|$ if $H-h$ contains no isolated vertex, where $h\in V(H)$. The main result of this paper is an improvement on this bound, which is most significant when $H$ is small or sparse, for instance when $H-h$ can be obtained from another graph by multiple edge subdivisions. Some consequences of this result are improvements on the upper bound for the cop number of $K_{3,t}$-minor-free graphs, $K_{2,t}$-minor-free graphs and linklessly embeddable graphs.

math.CO↗

arXiv2022

Sparsity of Graphs that Allow Two Distinct Eigenvalues

The parameter $q(G)$ of a graph $G$ is the minimum number of distinct eigenvalues over the family of symmetric matrices described by $G$. It is shown that the minimum number of edges necessary for a connected graph $G$ to have $q(G)=2$ is $2n-4$ if $n$ is even, and $2n-3$ if $n$ is odd. In addition, a characterization of graphs for which equality is achieved in either case is given.

math.CO↗

arXiv2022

An upper bound for the $k$-power domination number in $r$-uniform hypergraphs

Generalizing work on graphs, Chang and Roussel introduced $k$-power domination in hypergraphs and conjectured the upper bound for the $k$-power domination number for $r$-uniform hypergraphs on $n$ vertices was $\frac{n}{r+k}$. This upper bound was shown to be true for simple graphs ($r=2$) and it was further conjectured that only a family of hypergraphs, known as the squid hypergraphs, attained this upper bound. In this paper, the conjecture is proven to hold for hypergraphs with $r=3$ or $4$; but is shown to be false, by a counterexample, for $r\geq 7$. Furthermore, we show that the squid hypergraphs are not the only hypergraphs that attain the original upper bound. Finally, a new upper bound is proven for $r\geq 3$.

math.CO↗

arXiv2021

A Geometric Chung Lu model and the Drosophila Medulla connectome

Many real world graphs have edges correlated to the distance between them, but, in an inhomogeneous manner. While the Chung-Lu model and the geometric random graph models both are elegant in their simplicity, they are insufficient to capture the complexity of these networks. In this paper, we develop a generalized geometric random graph model that preserves many graph theoretic aspects of these real world networks. We test the validity of this model on a graphical representation of the Drosophila Medulla connectome.

math.CO↗

arXiv2019

Leaky Forcing: A New Variation of Zero Forcing

Zero forcing is a one-player game played on a graph. The player chooses some set of vertices to color, then iteratively applies a color change rule: If all but one of a colored vertex's neighbors are colored, color (i.e. "force") the remaining uncolored neighbor. Generally, the goal is to find the minimum number of vertices to initially color such that all vertices eventually become colored. Recently, equivalent formations of zero forcing have been developed in different settings including sensor allocation to solve linear systems (K.-Lin 2018), controllability in follower-leader dynamics (Monshizadeh-Zhang-Camlibel 2014), and edge covering in specific hypergraphs (Brimkov-Fast-Hicks 2016). While many variations of zero forcing are motivated by an associated minimum rank problem, these new formulations give new inspiration for new meaningful zero forcing variants. In our case, we study a new variation based on the linear algebraic interpretation mentioned above. In particular, what if there is a juncture in a network that has a leak, and, hence, is unreliable to facilitate solving a linear system on the network? In the context of zero forcing this corresponds to the following variation we call $\ell$-forcing: Given $\ell$, find a set of vertices such that for any set of $\ell$ vertices that are unable to force, all vertices will still be colored. We compute the $\ell$-forcing number for selected families of graphs including grid graphs. Perhaps surprisingly, we find examples where additional edges make the graph more "resilient" to these leaks. Further, we also implement known computational methods for our new leaky forcing variation.

math.CO↗

arXiv2019

Computing Bounds on Product-Graph Pebbling Numbers

Given a distribution of pebbles to the vertices of a graph, a pebbling move removes two pebbles from a single vertex and places a single pebble on an adjacent vertex. The pebbling number $π(G)$ is the smallest number such that, for any distribution of $π(G)$ pebbles to the vertices of $G$ and choice of root vertex $r$ of $G$, there exists a sequence of pebbling moves that places a pebble on $r$. Computing $π(G)$ is provably difficult, and recent methods for bounding $π(G)$ have proved computationally intractable, even for moderately sized graphs. Graham conjectured that $π(G ~\square~ H) \leq π(G) π(H)$, where $G ~\square~ H$ is the Cartesian product of $G$ and $H$ (1989). While the conjecture has been verified for specific families of graphs, in general it remains open. This study combines the focus of developing a computationally tractable, IP-based method for generating good bounds on $π(G ~\square~ H)$, with the goal of shedding light on Graham's conjecture.We provide computational results for a variety of Cartesian-product graphs, including some that are known to satisfy Graham's conjecture and some that are not. Our approach leads to a sizable improvement on the best known bound for $π(L ~\square~ L)$, where $L$ is the Lemke graph, and $L ~\square~ L$ is among the smallest known potential counterexamples to Graham's conjecture.

math.CO↗

arXiv2018

Leveraging local network communities to predict academic performance

For more than 20 years, social network analysis of student collaboration networks has focused on a student's centrality to predict academic performance. And even though a growing amount of sociological literature has supported that academic success is contagious, identifying central students in the network alone does not capture how peer interactions facilitate the spread of academic success throughout the network. Consequently, we propose novel predictors that treat academic success as a contagion by identifying a student's learning community, consisting of the peers that are most likely to influence a student's performance in a course. We evaluate the importance of these learning communities by predicting academic outcomes in an introductory college statistics course with 103 students. In particular, we observe that by including these learning community predictors, the resulting model is 68 times more likely to be the correct model than the current state-of-the-art centrality network models in the literature.

cs.SI↗

arXiv2018

Pebbling on Directed Graphs with Fixed Diameter

Pebbling is a game played on a graph. The single player is given a graph and a configuration of pebbles and may make pebbling moves by removing 2 pebbles from one vertex and placing one at an adjacent vertex to eventually have one pebble reach a predetermined vertex. The pebbling number, $π(G)$, is the minimum number of pebbles such that regardless of their exact configuration, the player can use pebbling moves to have a pebble reach any predetermined vertex. Previous work has related $π(G)$ to the diameter of $G$. Clarke, Hochberg, and Hurlbert demonstrated that every connected undirected graph on $n$ vertices with diameter 2 has $π(G) = n$ unless it belongs to an exceptional family of graphs, consisting of those that can be constructed in a specific manner; in which case $π(G) = n +1$. By generalizing a result of Chan and Godbole, Postle showed that for a graph with diameter $d$, $π(G) \le n 2^{\lceil \frac{d}{2} \rceil} (1+o_n(1))$. In this article, we continue this study relating pebbling and diameter with a focus on directed graphs. This leads to some surprising results. First, we show that in an oriented directed graph $G$ (in the sense that if $i \to j$ then we cannot have $j \to i$), it is indeed the case that if $G$ has diameter 2, $π(G) = n$ or $n + 1$, and if $π(G) = n+1$, the directed graph has a very particular structure. In the case of general directed graphs (that is, if $i \to j$, we may or may not have an arc $j \to i$) with diameter 2, we show that $π(G)$ can be as large as $\frac32 n + 1$, and further, this bound is sharp. More generally, we show that for general directed graphs, $π(G) \le 2^d n / d + f(d)$ where $f(d)$ is some function of only $d$.

math.CO↗

arXiv2018

Pebbling on Graph Products and other Binary Graph Constructions

Pebbling on graphs is a two-player game which involves repeatedly moving a pebble from one vertex to another by removing another pebble from the first vertex. The pebbling number $π(G)$ is the least number of pebbles required so that, regardless of the initial configuration of pebbles, a pebble can reach any vertex. Graham conjectured that the pebbling number for the cartesian product, $G \hspace{1mm}\square\hspace{1mm} H$, is bounded above by $π(G) π(H)$. We show that $π(G\hspace{1mm}\square\hspace{1mm} H) \le 2π(G) π(H)$ and, more sharply, that $π(G \hspace{1mm}\square\hspace{1mm} H) \le (π(G)+|G|) π(H)$. Furthermore, we provide similar results for other graph products and graph operations.

math.CO↗

arXiv2016

Interception in Distance-Vector Routing Networks

Despite the large effort devoted to cybersecurity research over the last decades, cyber intrusions and attacks are still increasing. With respect to routing networks, route hijacking has highlighted the need to reexamine the existing protocols that govern traffic routing. In particular, our pri- mary question is how the topology of a network affects the susceptibility of a routing protocol to endogenous route misdirection. In this paper we define and analyze an abstract model of traffic interception (i.e. eavesdropping) in distance-vector routing networks. Specifically, we study al- gorithms that measure the potential of groups of dishonest agents to divert traffic through their infrastructure under the constraint that messages must reach their intended destinations. We relate two variants of our model based on the allowed kinds of lies, define strategies for colluding agents, and prove optimality in special cases. In our main theorem we derive a provably optimal monitoring strategy for subsets of agents in which no two are adjacent, and we extend this strategy to the general case. Finally, we use our results to analyze the susceptibility of real and synthetic networks to endogenous traffic interception. In the Autonomous Systems (AS) graph of the United States, we show that compromising only 18 random nodes in the AS graph surprisingly captures 10% of all traffic paths in the network in expectation when a distance-vector routing protocol is in use.

cs.CR↗

arXiv2015

Lower Bounds on the Distance Domination Number of a Graph

For an integer $k \ge 1$, a (distance) $k$-dominating set of a connected graph $G$ is a set $S$ of vertices of $G$ such that every vertex of $V(G) \setminus S$ is at distance at most~$k$ from some vertex of $S$. The $k$-domination number, $γ_k(G)$, of $G$ is the minimum cardinality of a $k$-dominating set of $G$. In this paper, we establish lower bounds on the $k$-domination number of a graph in terms of its diameter, radius and girth. We prove that for connected graphs $G$ and $H$, $γ_k(G \times H) \ge γ_k(G) + γ_k(H) -1$, where $G \times H$ denotes the direct product of $G$ and $H$.

math.CO↗

arXiv2015

A linear k-fold Cheeger inequality

Given an undirected graph $G$, the classical Cheeger constant, $h_G$, measures the optimal partition of the vertices into 2 parts with relatively few edges between them based upon the sizes of the parts. The well-known Cheeger's inequality states that $2 λ_1 \le h_G \le \sqrt {2 λ_1}$ where $λ_1$ is the minimum nontrivial eigenvalue of the normalized Laplacian matrix. Recent work has generalized the concept of the Cheeger constant when partitioning the vertices of a graph into $k > 2$ parts. While there are several approaches, recent results have shown these higher-order Cheeger constants to be tightly controlled by $λ_{k-1}$, the $(k-1)$-th nontrivial eigenvalue, to within a quadratic factor. We present a new higher-order Cheeger inequality with several new perspectives. First, we use an alternative higher-order Cheeger constant which considers an "average case" approach. We show this measure is related to the average of the first $k-1$ nontrivial eigenvalues of the normalized Laplacian matrix. Further, using recent techniques, our results provide linear inequalities using the $\infty$-norms of the corresponding eigenvectors. Consequently, unlike previous results, this result is relevant even when $λ_{k-1} \to 1$.

math.CO↗

arXiv2014

Bounds for the Zero-Forcing Number of Graphs with Large Girth

We investigate the zero-forcing number for triangle-free graphs. We improve upon the trivial bound, $δ\le Z(G)$ where $δ$ is the minimum degree, in the triangle-free case. In particular, we show that $2 δ- 2 \le Z(G)$ for graphs with girth of at least 5, and this can be further improved when $G$ has a small cut set. Using these results, we are able to prove the Graph Complement Conjecture on minimum rank for a large class of graphs. Lastly, we make a conjecture that the lower bound for $Z(G)$ increases as a function of the girth, $g$, and $δ$.

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