Variants of Cop and Robber Problem

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  • Topic: Graph theory, Graph, Planar graph
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  • Published : December 8, 2012
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On Cop and Robber Games with a Tunneling Robber
Abhishek Parthasarathy Gautham Srinivas Mithraa Varun S Sreevatsan Vaidyanthan Sri Raghavan Prof.Nagarajan Krishnamurthy

Abstract Let G be a finite connected graph. Two players, called Cop C and Robber R, play a game on G according to the following rules.First C then R occupy some vertex of G. After that they move alternately along the edges of G. The cop C wins if he succeeds in putting himself on top of the robber R, otherwise R wins. Both C and R play rationally.In this paper we extend a few results shown in the paper written by Aigner and Fromme[1] and introduce the concept of building tunnels (connecting any two vertices in the graph) and analyse various cop-win and robber-win graphs. We also look into Intrusion detection games where the Cops can go to any vertex and the robbers can multiply.

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Introduction

Let G be a finite connected, undirected graph, with G(E,V), where E is the set of all edges and V is the set of all vertices. Let there be two players on the graph, such that player 1 is the Robber R and player 2 is the cop C, who play a game S on G according to the following rules: first the cop C and then the robber R choose vertices on which they position themselves. Then they in turn, alternatively along the edges of G. The cop player c wins if he ever succeeds in catching the robber, i.e, occupying the same vertex as the robber player R at some point in the game. Correspondingly, the robber player R wins if he manages to always evade the cop player, i.e., not occupying the same vertex as the cop player C. This game is a perfect information game where both the cop player c and the robber player R can see the entire graph and each others information. The game in the form as just described, i.e. with complete information on both sides, has also been studied by Aigner and Fromme[1], Nowakowski and Winkler[2], Quilliot[3] and possibly others. Let τ denote the set of cop win strategies on a given graph G. For each τ , there exists a c(G), which represents the cop number for that particular graph G. The cop number is defined as the minimum number of cops the cop player C requires inorder to make the graph G a cop win graph, i.e, a unique c(G) exists for every graph G which has a strategy belonging to τ . It is obvious that for every graph G one of the players must win, in fact, if C has a winning strategy,then he should succeed in catching R after at most n(n-1)+1 moves (n=number of vertices in G) since he can avoid repeated positions. Let us denote by C the class of cop-win graphs (i.e. those graphs on which C has a winning strategy) and by R the class of robber-win graphs. All graphs are assumed to be finite, undirected and connected.

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Examples

(For 1-cop 1-robber game) * Trees are cop-win graphs * Any complete graph is a cop-win graph as any vertex is connected to every other vertex. * Any cycle of length greater than or equal to 4 is a robber-win graph as at any point the robber can stay at least 2 vertices away from the cop. There are two different versions of the game namely the active version and the passive version.In the active version of the game the robber has to move at every turn. On the other hand in the passive version of the game the robber can chose not to move during his turn. The results in this paper are applicable to the passive version of the game.

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Definitions and Preliminaries
Pitfalls

A pitfall is defined as a vertex p, such that in a set (p,d), the dominating vertex d is connected to all the neighbours N(p) and p itself, i.e, N(p) U p N(d). thus, any vertex a robber R on p can go to is also reachable from d.

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Tunnel

A tunnel is defined as a graph construct the robber player can add to the given graph G(E,V) such that traversing from any vertex u V to any other vertex v V is possible. Essentially, a tunnel is analogous to a path between (u,v) of unit length.

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Tunnel Construction...
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