Variations of the cop and robber game on graphs
Espen Slettnes, Carl Joshua Quines, Shen-Fu Tsai, Jesse Geneson

TL;DR
This paper explores various theoretical aspects of the cop and robber game on graphs, including new game variants, graph conditions for winning, and algorithms for optimal pursuit strategies.
Contribution
It introduces a symmetric variation called the cop and killer game, characterizes graph conditions for winning, and develops a generalized Dijkstra's algorithm for pursuit-evasion problems.
Findings
Almost all random graphs are stalemate in the cop and killer game for certain edge probabilities.
Graphs with specific triangle and cycle structures determine winning strategies.
A generalized Dijkstra's algorithm computes minimal capture times and evasion probabilities.
Abstract
We prove new theoretical results about several variations of the cop and robber game on graphs. First, we consider a variation of the cop and robber game which is more symmetric called the cop and killer game. We prove for all that almost all random graphs are stalemate for the cop and killer game, where each edge occurs with probability such that . We prove that a graph can be killer-win if and only if it has exactly triangles or none at all. We prove that graphs with multiple cycles longer than triangles permit cop-win and killer-win graphs. For and , we show that there are cop-win and killer-win graphs with s. In addition, we identify game outcomes on specific graph products. Next, we find a generalized version of Dijkstra's algorithm that can be applied to find…
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Taxonomy
TopicsGame Theory and Applications · Experimental Behavioral Economics Studies · Spam and Phishing Detection
