Leiden University Student Repository

Bergh, M. van den (2017)

The game of Cops and Robbers on geometric graphs

Master thesis | Mathematics (MSc)

The Game of Cops and Robbers is played by two players on a graph, the one player controlling a set of cops and the other controlling a robber. The goal of the cops is to capture the robber in a finite number of moves, whereas the robber wants to evade the cops indefinitely. The central question we ask is: how many cops are needed on a given graph for them to be able to win the game? This amount is called the cop number of a graph. In this thesis, we give an extensive overview of results that give an upper bound on the cop number for graphs of a given type, like unit disk graphs and string graphs, as well as some examples of these graphs having a high cop number. In this overview, we will see that the results all greatly depend on the statement or proof method of one of two theorems published by Aigner and Fromme [1]. In addition to a survey of the existing work, we explore whether the two main theorems can be used to further sharpen these results. Leading to mostly negative answers,...Show moreThe Game of Cops and Robbers is played by two players on a graph, the one player controlling a set of cops and the other controlling a robber. The goal of the cops is to capture the robber in a finite number of moves, whereas the robber wants to evade the cops indefinitely. The central question we ask is: how many cops are needed on a given graph for them to be able to win the game? This amount is called the cop number of a graph. In this thesis, we give an extensive overview of results that give an upper bound on the cop number for graphs of a given type, like unit disk graphs and string graphs, as well as some examples of these graphs having a high cop number. In this overview, we will see that the results all greatly depend on the statement or proof method of one of two theorems published by Aigner and Fromme [1]. In addition to a survey of the existing work, we explore whether the two main theorems can be used to further sharpen these results. Leading to mostly negative answers, we conjecture that a different approach is necessary in order to book further progress.Show less

Author

Bergh, M. van den

Faculty

Faculty of Science

Supervisors

Spieksma, F.M.
Müller, T.

ECTS Credits

40

Language

en