Related papers: Cuts, Cats, and Complete Graphs

The sparsest cut problem consists of identifying a small set of edges that breaks the graph into balanced sets of vertices. The normalized cut problem balances the total degree, instead of the size, of the resulting sets. Applications of…

The semi-random graph process is a single player game in which the player is initially presented an empty graph on $n$ vertices. In each round, a vertex $u$ is presented to the player independently and uniformly at random. The player then…

There is a graph reduction system so that every optimal 1-planar graph can be reduced to an irreducible extended wheel graph, provided the reductions are applied such that the given graph class is preserved. A graph is optimal 1-planar if…

Cops and Robber is a well-studied two-player pursuit-evasion game played on a graph, where a group of cops tries to capture the robber. The \emph{cop number} of a graph is the minimum number of cops required to capture the robber.…

In this paper we consider positional games where the winning sets are tree universal graphs. Specifically, we show that in the unbiased Maker-Breaker game on the complete graph $K_n$, Maker has a strategy to occupy a graph which contains…

We propose a modification to the random destruction of graphs: Given a finite network with a distinguished set of sources and targets, remove (cut) vertices at random, discarding components that do not contain a source node. We investigate…

A new efficient algorithm is presented for finding all simple cycles that satisfy a length constraint in a directed graph. When the number of vertices is non-trivial, most cycle-finding problems are of practical interest for sparse graphs…

The game of cops and robbers is played on a fixed (finite or infinite) graph $G$. The cop chooses his starting position, then the robber chooses his. After that, they take turns and move to adjacent vertices, or stay at their current…

Wildlife corridors are components of landscapes, which facilitate the movement of organisms and processes between intact habitat areas, and thus provide connectivity between the habitats within the landscapes. Corridors are thus regions…

Consider a two-person zero-sum search game between a hider and a searcher. The hider hides among $n$ discrete locations, and the searcher successively visits individual locations until finding the hider. Known to both players, a search at…

In this article, we study the behavior of a broad family of real sequences derived from randomized one-pile subtraction games. For any subtraction set $S$, we allow any valid number of chips $s\in S$ to be removed at equal probability at…

We investigate Hunters & Rabbit game, where a set of hunters tries to catch an invisible rabbit that slides along the edges of a graph. We show that the minimum number of hunters required to win on an (n\times m)-grid is \lfloor…

We explore the Hunters and Rabbits game on the hypercube. In the process, we find the solution for all classes of graphs with an isoperimetric nesting property and find the exact hunter number of $Q^n$ to be $1+\sum\limits_{i=0}^{n-2}…

We introduce and study Maker/Breaker-type positional games on random graphs. Our main concern is to determine the threshold probability $p_{F}$ for the existence of Maker's strategy to claim a member of $F$ in the unbiased game played on…

We study a competitive optimization version of $\alpha'(G)$, the maximum size of a matching in a graph $G$. Players alternate adding edges of $G$ to a matching until it becomes a maximal matching. One player (Max) wants that matching to be…

In an introductory paper on dollar game played on a graph, Matt Baker wrote the following: ``The total number of borrowing moves required to win the game when playing the 'borrowing binge strategy' is independent of which borrowing moves…

This paper aims to solve the optimal strategy against a well-known adaptive algorithm, the Hedge algorithm, in a finitely repeated $2\times 2$ zero-sum game. In the literature, related theoretical results are very rare. To this end, we make…

The semi-random graph process is a single player game in which the player is initially presented an empty graph on $n$ vertices. In each round, a vertex $u$ is presented to the player independently and uniformly at random. The player then…

In 2016, Bonato, Janssen, and Roshanbin introduced graph burning as a discrete process that models the spread of social contagion. Although the burning process is a simple algorithm, the problem of determining the least number of rounds…

We obtain sharp bounds for the number of n-cycles in a finite graph as a function of the number of edges, and prove that the complete graph is optimal in more ways than could be imagined. En route, we prove some sharp estimates on power…