Abstract

Cops and Robbers is a vertex-pursuit game played on a graph where a set of cops attempts to capture a robber. Meyniel's Conjecture gives as asymptotic upper bound on the cop number, the number of cops required to win on a connected graph. The incidence graphs of affine planes meet this bound from below, they are called Meyniel extremal. The new parameters mқ and Mқ describe the minimum orders of k-cop-win graphs. The relation of these parameters to Meyniel's Conjecture is discussed. Further, the cop number for all connected graphs of order 10 or less is given. Finally, it is shown that cop win hypergraphs, a generalization of graphs, cannot be characterized in terms of retractions in the same manner as cop win graphs. This thesis presents some small steps towards a solution to Meyniel's Conjecture.

Highlights

  • Mathematics is a game with rules but no objectives”

  • We introduce so-called Meyniel extremal graphs which realize the upper bound in the conjecture, and give infinitely many new examples of such families satisfying certain regularity conditions

  • In this chapter we consider a different problem: How many k-cop-win graphs of order n exist? In this chapter we define two new parameters mk and Mk that describe the minimum orders of k-cop-win graphs

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Summary

Introduction to Cops and Robbers

A well-known, anonymous quote is “Philosophy is a game with objectives but no rules. Mathematics is a game with rules but no objectives”. Cops and Robbers is a game within Mathematics with both rules and objectives. Cops and Robbers is a vertex-pursuit game played on a graph for reasons that will be more clear as we give the rules for the game. Cop-win graphs, such as the graph, have a well known characterization based on the successive deletion of corners. This parameter is well-defined since placing a cop on every vertex will always result in a win (in one move) for the cops. Cops and Robbers belongs to a family of combinatorial games called vertex-pursuit games or graph searching games. These games provide a simplistic model for network security. We can think of the robber as an intruder in a network and cops as network monitors

Graphs
Incidence Graphs
Asymptotic Notation
Outline of Thesis
Steiner 2-designs
Projective Planes
Affine and partial affine planes
Introduction Meyniel’s
Minimum orders and growth rates
Algorithms for computing cop number
The k-cop-win Algorithm
Lower bounds
Introduction While the game Cops and
Conclusions and Future Work
Full Text
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