Related papers: The capture time of grids

We consider a variation of a cops and robbers game in which the cop---here referred to as "hunter"---is not constrained by the graph but must play in the dark against a "mole." We characterize the graphs---which we will call…

In this note, we prove that all cop-win graphs G in the game in which the robber and the cop move at different speeds s and s' with s'<s, are \delta-hyperbolic with \delta=O(s^2). We also show that the dependency between \delta and s is…

Meyniel's conjecture states that $n$-vertex connected graphs have cop number $O(\sqrt{n})$. The current best known upper bound is $n/2^{(1-o(1))\sqrt{\log n}}$, proved independently by Lu and Peng (2011), and by Scott and Sudakov (2011). In…

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…

Aigner and Fromme initiated the systematic study of the cop number of a graph by proving the elegant and sharp result that in every connected planar graph, three cops are sufficient to win a natural pursuit game against a single robber.…

The cop throttling number of a graph, introduced in 2018 by Breen et al., optimizes the balance between the number of cops used and the number of rounds required to catch the robber in a game of Cops and Robbers. In 2019, Cox and Sanaei…

A recent paper by Bhatia, Chin, Mani, and Mossel (2026) defined stochastic processes aimed at modeling the game of War for {\em two players} with $n$ cards. That paper showed that these models, assuming uniform random decks, are equivalent…

We present two efficient algorithms that compute the optimal strategy for cop in the game of Cop v.s. Gambler where the gambler's strategy is not optimal but known to the cop. The first algorithm is analogous to Bellman-Ford algorithm for…

The cop number of a graph $G$ is the smallest $k$ such that $k$ cops win the game of cops and robber on $G$. We investigate the maximum cop number of geometric intersection graphs, which are graphs whose vertices are represented by…

In chomp on graphs, two players alternatingly pick an edge or a vertex from a graph. The player that cannot move any more loses. The questions one wants to answer for a given graph are: Which player has a winning strategy? Can a explicit…

We consider a surrounding variant of cops and robbers on graphs of bounded genus. We obtain bounds on the number of cops required to surround a robber on planar graphs, toroidal graphs, and outerplanar graphs. We also obtain improved bounds…

In the game of cops and robber, the cops try to capture a robber moving on the vertices of the graph. The minimum number of cops required to win on a given graph $G$ is called the cop number of $G$. The biggest open conjecture in this area…

The traditional game of cops and robbers is played on undirected graph. Recently, the same game played on directed graph is getting attention by more and more people. We knew that if we forbid some subgraph we can bound the cop number of…

By the Grid Minor Theorem of Robertson and Seymour, every graph of sufficiently large tree-width contains a large grid as a minor. Tree-width may therefore be regarded as a measure of 'grid-likeness' of a graph. The grid contains a long…

In the $(s,d)$-spy game over a graph, introduced by Cohen et al. in 2016, one spy and $k$ guards occupy vertices of a graph and, at each turn, each guard may move along one edge and the spy may move along at most $s$ edges. The guards win…

We consider a game with two piles, in which two players take turn to add $a$ or $b$ chips ($a$, $b$ are not necessarily positive) randomly and independently to their respective piles. The player who collects $n$ chips first wins the game.…

We apply the power-of-two-choices paradigm to a random walk on a graph: rather than moving to a uniform random neighbour at each step, a controller is allowed to choose from two independent uniform random neighbours. We prove that this…

A \emph{periodic graph} ${\cal G}=(G_0, G_1, G_2, \dots)$ with period $p$ is an infinite periodic sequence of graphs $G_i = G_{i + p} = (V,E_i)$, where $i \geq 0$. The graph $G=(V,\cup_i E_i)$ is called the footprint of ${\cal G}$.…

The game subset take-away begins with a simplicial complex \Delta. Two players take turns removing any element of \Delta as well as all other elements which contain it, and the last player able to move wins. Graph Chomp is a special case of…

We provide a game theoretic framework for the game of cops and robbers (CR). Within this framework we study certain assumptions which underlie the concepts of optimal strategies and capture time. We also point out a connection of these…