Related papers: Cops and Robbers on Intersection Graphs
This paper describes a 720-vertex connected planar graph G such that cop1(G), denoting the minimum number of cops needed to catch the robber in the 1-cop-move game on G, is at least 4 and at most 7. Furthermore, G has a connected subgraph H…
We consider a variant of Cops and Robbers wherein each edge traversed by the robber is deleted from the graph. The focus is on determining the minimum number of cops needed to capture a robber on a graph $G$, called the {\em bridge-burning…
We consider a variant of Cops and Robbers in which both the cops and the robber are allowed to traverse up to $s$ edges on each of their turns, where $s \ge 2$. We give several general for this new model as well as establish bounds for the…
We prove that the cop number of any $2K_2$-free graph is at most 2, proving a conjecture of Sivaraman and Testa. We also show that the upper bound of $3$ on the cop number of $2K_1+K_2$-free (co-diamond--free) graphs is best possible.
A hole in a graph is an induced cycle of length at least 4. We give a simple winning strategy for t-3 cops to capture a robber in the game of cops and robbers played in a graph that does not contain a hole of length at least t. This…
The game of cops and robbers, played on a fixed graph $G$, is a two-player game, where the cop and the robber (the players) take turns in moving to adjacent vertices. The game finishes if the cop lands on the robber's vertex. In that case…
We consider a cops and robber game where the cops are blocking edges of a graph, while the robber occupies its vertices. At each round of the game, the cops choose some set of edges to block and right after the robber is obliged to move to…
We prove that the cop number of a $2K_2$-free graph is at most $2$ if it has diameter $3$ or does not have an induced cycle of length $k$, where $k \ \in \{3,4,5\}$. We conjecture that the cop number of every $2K_2$-free graph is at most…
We compare two kinds of pursuit-evasion games played on graphs. In Cops and Robbers, the cops can move strategically to adjacent vertices as they please, while in a new variant, called deterministic Zombies and Survivors, the zombies (the…
The game of cops and robber is a pursuit-evasion game played on graphs that has been extensively studied. Traditionally the game is played with one or more cops and only one robber, but in this paper we consider the game played with…
Given finitely many connected polygonal obstacles $O_1,\dots,O_k$ in the plane and a set $P$ of points in general position and not in any obstacle, the {\em visibility graph} of $P$ with obstacles $O_1,\dots,O_k$ is the (geometric) graph…
The main goal of this paper is to formalize and explore a connection between chromatic properties of graphs with geometric representations and competitive analysis of on-line algorithms, which became apparent after the recent construction…
Cops and robbers is a vertex-pursuit game played on graphs. In the classical cops-and-robbers game, a set of cops and a robber occupy the vertices of the graph and move alternately along the graph's edges with perfect information about each…
In this paper, we relate the problem of finding a maximum clique to the intersection number of the input graph (i.e. the minimum number of cliques needed to edge cover the graph). In particular, we consider the maximum clique problem for…
The interval number of a graph $G$ is the minimum $k$ such that one can assign to each vertex of $G$ a union of $k$ intervals on the real line, such that $G$ is the intersection graph of these sets, i.e., two vertices are adjacent in $G$ if…
In the game of Cops and Robbers, one of the most useful results is that an isometric path in a graph can be guarded by one cop. In this paper, we introduce the concept of wide shadow in a subgraph, and use it to characterize all 1-guardable…
We investigate a cops and robber game on directed graphs, where the robber moves along the arcs of the graph, while the cops can select any position at each time step. Our main focus is on the cop number: the minimum number of cops required…
We investigate a pursuit-evasion game on an undirected graph in which a robber, moving at a fixed constant speed, attempts to evade a team of cops who are blind to the robber's location and can quickly travel between any pair of vertices in…
A generalized Petersen graph $GP(n,k)$ is a regular cubic graph on $2n$ vertices (the parameter $k$ is used to define some of the edges). It was previously shown (Ball et al., 2015) that the cop number of $GP(n,k)$ is at most $4$, for all…
The game of Cops and Robbers is a pursuit-evasion game on graphs that has been extensively studied in finite settings, particularly through the concept of cop number. In this paper, we explore infinite variants of the game, focusing on the…