Variants of Cop and Robber Problem
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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1
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
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.
i
1
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