Franklin Kenter

Note on Power Propagation Time and Lower Bounds for the Power Domination Number

We present a counterexample to a lower bound for the power domination number given in Liao (J Comb Optim 31:725–742, 2016). We also define the power propagation time, using the power domination propagation ideas in Liao and the (zero forcing) propagation time in Hogben et al. (Discrete Appl Math 160:1994–2005, 2012).

How to Make the Perfect Fireworks Display: Two Strategies for Hanabi

Summary The game of Hanabi is a multiplayer cooperative card game that has many similarities to a mathematical “hat guessing game.” In Hanabi, a player does not see the cards in her own hand and must rely on the actions of the other players to determine information about her cards. This article presents two strategies for Hanabi. These strategies use different encoding schemes, based on ideas from network coding, to efficiently relay information. The first strategy allows players to effectively recommend moves for other players, and the second strategy allows players to determine the contents of their hands. Results from computer simulations demonstrate that both strategies perform well. In particular, the second strategy achieves a perfect score more than 75 percent of the time.

The relationship between k-forcing and k-power domination

Abstract Zero forcing and power domination are iterative processes on graphs where an initial set of vertices are observed, and additional vertices become observed based on some rules. In both cases, the goal is to eventually observe the entire graph using the fewest number of initial vertices. The concept of k -power domination was introduced by Chang et al. (2012) as a generalization of power domination and standard graph domination. Independently, k -forcing was defined by Amos et al. (2015) to generalize zero forcing. In this paper, we combine the study of k -forcing and k -power domination, providing a new approach to analyze both processes. We give a relationship between the k -forcing and the k -power domination numbers of a graph that bounds one in terms of the other. We also obtain results using the contraction of subgraphs that allow the parallel computation of k -forcing and k -power dominating sets.

Approximating the minimum rank of a graph via alternating projection

The minimum rank problem asks to find the minimum rank over all matrices with a given pattern associated with a graph. This problem is NP-hard, and there is no known approximation method. Further, this problem has no straightforward convex relaxation. In this article, a numerical algorithm is given to heuristically approximate the minimum rank using alternating projections. The effectiveness of this algorithm is demonstrated by comparing its results to a related parameter: the zero-forcing number. Using these methods, numerical evidence for the minimum rank graph complement conjecture is provided.

Iterated prisoner's dilemma with extortionate zero-determinant strategies and random-memory opponents

We investigate Extortionate Zero Determinant (EZD) strategies for the iterated prisoners dilemma (IPD) against random memory-based strategies without mutation. These strategies are randomly-generated deterministic instructions with a predetermined average level of cooperation. While EZD strategies generally outperform any single evolutionary opponent, we show that under certain conditions EZD strategies can be taken over by a small number of randomly-generated deterministic strategies and force the EZD to extinction. We demonstrate that a major determining factor contributing to this phenomena is the maximum score among all non-EZD strategies when playing against itself. In contrast, we theoretically analyze the IPD consisting of an EZD strategy and a purely random one that cooperates each round with a predetermined probability. In this latter case, the EZD almost always prevails though the purely random player survives. In this sense, we conclude that even without mutation, it is still evolutionarily advantageous for the IPD to have a diverse initial random population.

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.

On the Principal Permanent Rank Characteristic Sequences of Graphs and Digraphs

The principal permanent rank characteristic sequence is a binary sequence

Bounds for the Zero Forcing Number of Graphs with Large Girth

r_0 r_1 \ldots r_n

On the Distance Spectra of Graphs

where

Discrepancy inequalities for directed graphs

r_k = 1