arXiv subjects

Melissa A. Huggan

Publications and source records attributed to Melissa A. Huggan.

At least 19 recordsLinked to original sources

arXiv2026

The Infectious Vaccination Problem: a variant of Firefighting with Spreading Defence

The Firefighter Problem models a spreading process (originally a fire, alternatively an infection or rumour, for example) on a graph. A defender saves a single vertex per turn; after each defence, the fire spreads to the unburned and undefended neighbours of all burning vertices. Deciding whether a strategy exists for the defender to protect some targeted number of vertices is computationally hard in graphs in general, but tractable in some restricted cases. Inspired by research into spreadable rabies vaccines for bats, we study a variant of the Firefighter problem in which defence also spreads. Some approximation results are already known for this problem; we provide algorithmic and hardness results, as well as containment results for the infinite $n$-dimensional Cartesian and strong grid graphs.

cs.DS↗

arXiv2025

Minimal Graph Embeddings via Point Deletions in Steiner triple systems

The game Nofil is a two-player combinatorial game in which players take turns marking points of a design such that the set of marked points does not contain a block. Equivalently, we can think of the points as being deleted from the design and points that are on singleton sets can no longer be marked. Every game play eventually results in the design becoming a graph. Previous work has shown that every graph is reachable from some Steiner triple system (STS), although the order of the constructed STS is often far from the known lower bounds. In this paper we give embeddings of complete graphs and star graphs into a $\STS$ that is minimal or very nearly meets the bounds. We further discuss possible minimal embeddings of empty graphs, paths, and cycles.

math.CO↗

arXiv2025

Deduction with $k$ moves

The deduction game may be thought of as a variant on the classical game of cops and robber in which the cops (searchers) aim to capture an invisible robber (evader); each cop is allowed to move at most once, and cops situated on different vertices cannot communicate to co-ordinate their strategy. In this paper, we extend the deduction game to allow each searcher to make $k$ moves, where $k$ is a fixed positive integer. We consider the value of the $k$-move deduction number on several classes of graphs including paths, cycles, complete graphs, complete bipartite graphs, and Cartesian and strong products of paths.

math.CO↗

arXiv2025

A note on the strength of a hypercube

As a generalization of super magic strength, the strength of a graph was introduced in [R. Ichishima, F.A. Muntaner-Batle, A. Oshima, Bounds for the strength of graphs, Austral. J. of Combin. 72(3) (2018) 492-508]. For a vertex ordering $f$ of graph $G$, the strength of $f$ is the maximum sum of the labels on any pair of adjacent vertices. The strength of $G$ is defined as the minimum strength of $f$, taken over all vertex orderings of $G$. The strength of the hypercube is unknown, but bounded. In this note, we provide an improved upper bound for the strength of a hypercube.

math.CO↗

arXiv2024

The damage number of the Cartesian product of graphs

We consider a variation of Cops and Robber, introduced in [D. Cox and A. Sanaei, The damage number of a graph, [Aust. J. of Comb. 75(1) (2019) 1-16] where vertices visited by a robber are considered damaged and a single cop aims to minimize the number of distinct vertices damaged by a robber. Motivated by the interesting relationships that often emerge between input graphs and their Cartesian product, we study the damage number of the Cartesian product of graphs. We provide a general upper bound and consider the damage number of the product of two trees or cycles. We also consider graphs with small damage number.

math.CO↗

arXiv2024

Cops and Attacking Robbers with Cycle Constraints

This paper considers the Cops and Attacking Robbers game, a variant of Cops and Robbers, where the robber is empowered to attack a cop in the same way a cop can capture the robber. In a graph $G$, the number of cops required to capture a robber in the Cops and Attacking Robbers game is denoted by $\attCop(G)$. We characterise the triangle-free graphs $G$ with $\attCop(G) \leq 2$ via a natural generalisation of the cop-win characterisation by Nowakowski and Winkler \cite{nowakowski1983vertex}. We also prove that all bipartite planar graphs $G$ have $\attCop(G) \leq 4$ and show this is tight by constructing a bipartite planar graph $G$ with $\attCop(G) = 4$. Finally we construct $17$ non-isomorphic graphs $H$ of order $58$ with $\attCop(H) = 6$ and $\cop(H)=3$. This provides the first example of a graph $H$ with $\attCop(H) - \cop(H) \geq 3$ extending work by Bonato, Finbow, Gordinowicz, Haidar, Kinnersley, Mitsche, Prałat, and Stacho \cite{bonato2014robber}. We conclude with a list of conjectures and open problems.

math.CO↗

arXiv2024

Tic-Tac-Toe on Designs

We consider playing the game of Tic-Tac-Toe on block designs BIBD($v, k, λ$) and transversal designs TD($k, n$). Players take turns choosing points and the first player to complete a block wins the game. We show that triple systems, BIBD($v, 3, λ$), are a first player win if and only if $v \geq 5$. Further, we show that for $k = 2, 3$, TD($k, n$) is a first player win if and only if $n \geq k$. We also consider a weak version of the game, called Maker-Breaker, in which the second player wins if they can stop the first player from winning. In this case, we adapt known bounds for when either the first or second player can win on BIBD($v, k, 1$) and TD($k, n$), and show that for Maker-Breaker, BIBD($v, 4, 1$) is a first player win if and only if $v \geq 16$. We show that TD($4, 4$) is a second player win, and so the second player can force a draw in the regular game by playing the same strategy.

math.CO↗

arXiv2022

Cheating Robot Games: A model for insider information

Combinatorial games are two-player games of pure strategy where the players, usually called Left and Right, move alternately. In this paper, we introduce Cheating Robot games. These arise from simultaneous-play combinatorial games where one player has insider information ('cheats'). Play occurs in rounds. At the beginning of a round, both players know the moves that are available to them. Left chooses a move. Knowing Left's move, Right then chooses a move. Right's move is not constrained by Left's choice. The round is not completed until both players have made a choice. A game is finished only when one or both players do not have a move at the beginning of a round. Right choosing a move, knowing Left's, makes the games deterministic, distinguishing them from simultaneous games. Also, the ending condition distinguishes this class of games from combinatorial games, since the outcomes are now Left-win, Right-win and draw. The basic theory and properties are developed, including showing that there is an equivalence relation and partial order on the games. Whilst there are no inverses in the class of all games, we show that there is a sub-class, simple hot games, in which the integers have inverses. In this sub-class, the optimal strategies are obtained by the solutions to a minimum-weight matching problem on a graph whose number of vertices equals the number of summands in the disjunctive sum.

math.CO↗

arXiv2022

Tic-Tac-Toe on an Affine Plane of order 4

The game of tic-tac-toe is well known. In particular, in its classic version it is famous for being unwinnable by either player. While classically it is played on a grid, it is natural to consider the effect of playing the game on richer structures, such as finite planes. Playing the game of tic-tac-toe on finite affine and projective planes has been studied previously. While the second player can usually force a draw, for small orders it is possible for the first player to win. In this regard, a computer proof that tic-tac-toe played on the affine plane of order 4 is a first player win has been claimed. In this note we use techniques from the theory of latin squares and transversal designs to give a human verifiable, explicit proof of this fact.

math.CO↗

arXiv2021

The localization capture time of a graph

The localization game is a pursuit-evasion game analogous to Cops and Robbers, where the robber is invisible and the cops send distance probes in an attempt to identify the location of the robber. We present a novel graph parameter called the capture time, which measures how long the localization game lasts assuming optimal play. We conjecture that the capture time is linear in the order of the graph, and show that the conjecture holds for graph families such as trees and interval graphs. We study bounds on the capture time for trees and its monotone property on induced subgraphs of trees and more general graphs. We give upper bounds for the capture time on the incidence graphs of projective planes. We finish with new bounds on the localization number and capture time using treewidth.

math.CO↗

arXiv2021

The combinatorial game nofil played on Steiner Triple Systems

We introduce an impartial combinatorial game on Steiner triple systems called Nofil. Players move alternately, choosing points of the triple system. If a player is forced to fill a block on their turn, they lose. We explore the play of Nofil on all Steiner triple systems up to order 15 and a sampling for orders 19, 21, and 25. We determine the optimal strategies by computing the nim-values for each game and its subgames. The game Nofil can be thought of in terms of play on a corresponding hypergraph. As game play progresses, the hypergraph shrinks and will eventually be equivalent to playing the game Node Kayles on an isomorphic graph. Node Kayles is well studied and understood. Motivated by this, we study which Node Kayles positions can be reached, i.e. embedded into a Steiner triple system. We prove necessary conditions and sufficient conditions for the existence of such graph embeddings and conclude that the complexity of determining the outcome of the game Nofil on Steiner triple systems is PSPACE-complete.

math.CO↗

arXiv2021

Progress on the localization number of a graph

We highlight new results on the localization number of a graph, a parameter derived from the localization graph searching game. After introducing the game and providing an overview of existing results, we describe recent results on the localization number. We describe bounds or exact values of the localization number of incidence graphs of designs, polarity graphs, and Kneser graphs.

math.CO↗

arXiv2021

Cops and an Insightful Robber

The 'Cheating Robot' version of Cops and Robbers is played on a finite, simple, connected graph. The players move in the same time period. However, before moving, the robot observes to which vertices the cops are moving and it is fast enough to complete its move in the time period. The cops also know that the robot will use this information. More cops are required to capture a robot than to capture a robber. Indeed, the minimum degree is a lower bound on the number of cops required to capture a robot. Only on a tree is one cop guaranteed to capture a robot, although two cops are sufficient to capture both a robber and a robot on outerplanar graphs. In graphs where retracts are involved, we show how cop strategies against a robber can be modified to capture a robot. This approach gives exact numbers for hypercubes, and $k$-dimensional grids in general.

math.CO↗

arXiv2021

The game of Flipping Coins

We consider Flipping Coins, a partizan version of the impartial game Turning Turtles, played on lines of coins. We show the values of this game are numbers, and these are found by first applying a reduction, then decomposing the position into an iterated ordinal sum. This is unusual since moves in the middle of the line do not eliminate the rest of the line. Moreover, when $G$ is decomposed into lines $H$ and $K$, then $G=(H:K^R)$. This is in contrast to Hackenbush Strings where $G= (H:K)$.

math.CO↗

arXiv2021

An investigation into the application of genetic programming to combinatorial game theory

Genetic programming is the practice of evolving formulas using crossover and mutation of genes representing functional operations. Motivated by genetic evolution we develop and solve two combinatorial games, and we demonstrate some advantages and pitfalls of using genetic programming to investigate Grundy values. We conclude by investigating a combinatorial game whose ruleset and starting positions are inspired by genetic structures.

math.CO↗

arXiv2021

The iterated local transitivity model for hypergraphs

Complex networks are pervasive in the real world, capturing dyadic interactions between pairs of vertices, and a large corpus has emerged on their mining and modeling. However, many phenomena are comprised of polyadic interactions between more than two vertices. Such complex hypergraphs range from emails among groups of individuals, scholarly collaboration, or joint interactions of proteins in living cells. A key generative principle within social and other complex networks is transitivity, where friends of friends are more likely friends. The previously proposed Iterated Local Transitivity (ILT) model incorporated transitivity as an evolutionary mechanism. The ILT model provably satisfies many observed properties of social networks, such as densification, low average distances, and high clustering coefficients. We propose a new, generative model for complex hypergraphs based on transitivity, called the Iterated Local Transitivity Hypergraph (or ILTH) model. In ILTH, we iteratively apply the principle of transitivity to form new hypergraphs. The resulting model generates hypergraphs simulating properties observed in real-world complex hypergraphs, such as densification and low average distances. We consider properties unique to hypergraphs not captured by their 2-section. We show that certain motifs, which are specified subhypergraphs of small order, have faster growth rates in ILTH hypergraphs than in random hypergraphs with the same order and expected average degree. We show that the graphs admitting a homomorphism into the 2-section of the initial hypergraph appear as induced subgraphs in the 2-section of ILTH hypergraphs. We consider new and existing hypergraph clustering coefficients, and show that these coefficients have larger values in ILTH hypergraphs than in comparable random hypergraphs.

cs.DM↗

arXiv2021

A Note on Numbers

When are all positions of a game numbers? We show that two properties are necessary and sufficient. These properties are consequences of that, in a number, it is not an advantage to be the first player. One of these properties implies the other. However, checking for one or the other, rather than just one, can often be accomplished by only looking at the positions on the `board'. If the stronger property holds for all positions, then the values are integers.

math.CO↗

arXiv2020

Ordinal Sums, Clockwise Hackenbush, and Domino Shave

We present two rulesets, Domino Shave and Clockwise Hackenbush. The first is somehow natural and, as special cases, includes Stirling Shave and Hetyei's Bernoulli game. Clockwise Hackenbush seems artificial yet it is equivalent to Domino Shave. From the pictorial form of the game, and a knowledge of Hackenbush, the decomposition into ordinal sums is immediate. The values of Clockwise Blue-Red Hackenbush are numbers and we provide an explicit formula for the ordinal sum of numbers where the literal form of the base is $\{x\,|\,\}$ or $\{\,|\,x\}$, and $x$ is a number. That formula generalizes van Roode's signed binary number method for Blue-Red Hackenbush.

math.CO↗