![Page 1: The Game of Cops and Robbers on Graphs - Inria · 2017-11-06 · 1/19 The Game of Cops and Robbers on Graphs Fionn Mc Inerney Universit e C^ote d’Azur, Inria, CNRS, I3S, France](https://reader034.vdocuments.us/reader034/viewer/2022042119/5e97c5c04161dd2ff77594ab/html5/thumbnails/1.jpg)
1/19
The Game of Cops and Robbers on Graphs
Fionn Mc Inerney
Universite Cote d’Azur, Inria, CNRS, I3S, France
PhD Seminars of Inria Sophia-Antipolis - Mediterranee
November 6, 2017
(Modified Slides of Nicolas Nisse)
F. Mc Inerney The Game of Cops and Robbers on Graphs
![Page 2: The Game of Cops and Robbers on Graphs - Inria · 2017-11-06 · 1/19 The Game of Cops and Robbers on Graphs Fionn Mc Inerney Universit e C^ote d’Azur, Inria, CNRS, I3S, France](https://reader034.vdocuments.us/reader034/viewer/2022042119/5e97c5c04161dd2ff77594ab/html5/thumbnails/2.jpg)
2/19
What is a Graph?
A graph G
A set of vertices (dots) V (G ) connected by a set of edges(lines) E (G ).
Graphs model real-life networks: social (Facebook, Twitter),biological (food chains, protein-interaction networks),transportation (metro maps, bus routes), etc.
F. Mc Inerney The Game of Cops and Robbers on Graphs
![Page 3: The Game of Cops and Robbers on Graphs - Inria · 2017-11-06 · 1/19 The Game of Cops and Robbers on Graphs Fionn Mc Inerney Universit e C^ote d’Azur, Inria, CNRS, I3S, France](https://reader034.vdocuments.us/reader034/viewer/2022042119/5e97c5c04161dd2ff77594ab/html5/thumbnails/3.jpg)
3/19
What is a Graph?
A graph G
A set of vertices (dots) V (G ) connected by a set of edges(lines) E (G ).
Graphs model real-life networks: social (Facebook, Twitter),biological (food chains, protein-interaction networks),transportation (metro maps, bus routes), etc.
F. Mc Inerney The Game of Cops and Robbers on Graphs
![Page 4: The Game of Cops and Robbers on Graphs - Inria · 2017-11-06 · 1/19 The Game of Cops and Robbers on Graphs Fionn Mc Inerney Universit e C^ote d’Azur, Inria, CNRS, I3S, France](https://reader034.vdocuments.us/reader034/viewer/2022042119/5e97c5c04161dd2ff77594ab/html5/thumbnails/4.jpg)
4/19
Basic Graph Theory
Adjacency
Two vertices u and v are adjacent if they are connected by anedge.
N(v): Open-neighbourhood of v
The set of vertices adjacent to v .
N[v ]: Closed-neighbourhood of v
The set of vertices adjacent to v and v itself.
deg(v): Degree of v
|N(v)| or the number of distinct vertices adjacent to v .
F. Mc Inerney The Game of Cops and Robbers on Graphs
![Page 5: The Game of Cops and Robbers on Graphs - Inria · 2017-11-06 · 1/19 The Game of Cops and Robbers on Graphs Fionn Mc Inerney Universit e C^ote d’Azur, Inria, CNRS, I3S, France](https://reader034.vdocuments.us/reader034/viewer/2022042119/5e97c5c04161dd2ff77594ab/html5/thumbnails/5.jpg)
5/19
Basic Graph Theory Continued
d(u, v): Distance between u and v
The length of the shortest path between u and v .
γ(G ): Domination number of a graph G
The minimum size of a set S of vertices such that all verticesin G are either in S or adjacent to a vertex in S .
v2 v3 v4 v5
v1
v6 v7 v8
v3 v4
v6 v7 v8
v3
v3 is adjacent tov4, v6, v7, v8.
N(v3) = v4, v6, v7, v8deg(v3) = 4
N[v3] = v3, v4, v6, v7, v8
F. Mc Inerney The Game of Cops and Robbers on Graphs
![Page 6: The Game of Cops and Robbers on Graphs - Inria · 2017-11-06 · 1/19 The Game of Cops and Robbers on Graphs Fionn Mc Inerney Universit e C^ote d’Azur, Inria, CNRS, I3S, France](https://reader034.vdocuments.us/reader034/viewer/2022042119/5e97c5c04161dd2ff77594ab/html5/thumbnails/6.jpg)
5/19
Basic Graph Theory Continued
d(u, v): Distance between u and v
The length of the shortest path between u and v .
γ(G ): Domination number of a graph G
The minimum size of a set S of vertices such that all verticesin G are either in S or adjacent to a vertex in S .
v2 v3 v4 v5
v1
v6 v7 v8
v3
v4
v6 v7 v8
v3
v3 is adjacent tov4, v6, v7, v8.
N(v3) = v4, v6, v7, v8deg(v3) = 4
N[v3] = v3, v4, v6, v7, v8
F. Mc Inerney The Game of Cops and Robbers on Graphs
![Page 7: The Game of Cops and Robbers on Graphs - Inria · 2017-11-06 · 1/19 The Game of Cops and Robbers on Graphs Fionn Mc Inerney Universit e C^ote d’Azur, Inria, CNRS, I3S, France](https://reader034.vdocuments.us/reader034/viewer/2022042119/5e97c5c04161dd2ff77594ab/html5/thumbnails/7.jpg)
5/19
Basic Graph Theory Continued
d(u, v): Distance between u and v
The length of the shortest path between u and v .
γ(G ): Domination number of a graph G
The minimum size of a set S of vertices such that all verticesin G are either in S or adjacent to a vertex in S .
v2 v3 v4 v5
v1
v6 v7 v8
v3 v4
v6 v7 v8
v3
v3 is adjacent tov4, v6, v7, v8.
N(v3) = v4, v6, v7, v8deg(v3) = 4
N[v3] = v3, v4, v6, v7, v8
F. Mc Inerney The Game of Cops and Robbers on Graphs
![Page 8: The Game of Cops and Robbers on Graphs - Inria · 2017-11-06 · 1/19 The Game of Cops and Robbers on Graphs Fionn Mc Inerney Universit e C^ote d’Azur, Inria, CNRS, I3S, France](https://reader034.vdocuments.us/reader034/viewer/2022042119/5e97c5c04161dd2ff77594ab/html5/thumbnails/8.jpg)
5/19
Basic Graph Theory Continued
d(u, v): Distance between u and v
The length of the shortest path between u and v .
γ(G ): Domination number of a graph G
The minimum size of a set S of vertices such that all verticesin G are either in S or adjacent to a vertex in S .
v2 v3 v4 v5
v1
v6 v7 v8
v3
v4
v6 v7 v8
v3
v3 is adjacent tov4, v6, v7, v8.
N(v3) = v4, v6, v7, v8deg(v3) = 4
N[v3] = v3, v4, v6, v7, v8
F. Mc Inerney The Game of Cops and Robbers on Graphs
![Page 9: The Game of Cops and Robbers on Graphs - Inria · 2017-11-06 · 1/19 The Game of Cops and Robbers on Graphs Fionn Mc Inerney Universit e C^ote d’Azur, Inria, CNRS, I3S, France](https://reader034.vdocuments.us/reader034/viewer/2022042119/5e97c5c04161dd2ff77594ab/html5/thumbnails/9.jpg)
6/19
Basic Graph Theory Continued
d(u, v): Distance between u and v
The length of the shortest path between u and v .
γ(G ): Domination number of a graph G
The minimum size of a set S of vertices such that all verticesin G are either in S or adjacent to a vertex in S .
v2 v3 v4 v5
v1
v6 v7 v8
v2 v5
v1
v3
d(v2, v5) = 2
γ(G ) = 2
F. Mc Inerney The Game of Cops and Robbers on Graphs
![Page 10: The Game of Cops and Robbers on Graphs - Inria · 2017-11-06 · 1/19 The Game of Cops and Robbers on Graphs Fionn Mc Inerney Universit e C^ote d’Azur, Inria, CNRS, I3S, France](https://reader034.vdocuments.us/reader034/viewer/2022042119/5e97c5c04161dd2ff77594ab/html5/thumbnails/10.jpg)
6/19
Basic Graph Theory Continued
d(u, v): Distance between u and v
The length of the shortest path between u and v .
γ(G ): Domination number of a graph G
The minimum size of a set S of vertices such that all verticesin G are either in S or adjacent to a vertex in S .
v2 v3 v4 v5
v1
v6 v7 v8
v2 v5
v1
v3
d(v2, v5) = 2
γ(G ) = 2
F. Mc Inerney The Game of Cops and Robbers on Graphs
![Page 11: The Game of Cops and Robbers on Graphs - Inria · 2017-11-06 · 1/19 The Game of Cops and Robbers on Graphs Fionn Mc Inerney Universit e C^ote d’Azur, Inria, CNRS, I3S, France](https://reader034.vdocuments.us/reader034/viewer/2022042119/5e97c5c04161dd2ff77594ab/html5/thumbnails/11.jpg)
7/19
Pursuit-Evasion Games
2-Player games
A team of mobile entities (Cops) track down another mobile entity (Robber)
Always one winner
Combinatorial Problem:Minimizing some resource for some Player to wine.g., minimize number of Cops to capture the Robber.
Algorithmic Problem:Computing winning strategy (sequence of moves) for some Playere.g., compute strategy for Cops to capture Robber/Robber to avoid capture.
natural applications: coordination of mobile autonomous agents(Robotics, Network Security, Information Seeking...)
but also: Graph Theory, Models of Computation, Logic, Routing...
F. Mc Inerney The Game of Cops and Robbers on Graphs
![Page 12: The Game of Cops and Robbers on Graphs - Inria · 2017-11-06 · 1/19 The Game of Cops and Robbers on Graphs Fionn Mc Inerney Universit e C^ote d’Azur, Inria, CNRS, I3S, France](https://reader034.vdocuments.us/reader034/viewer/2022042119/5e97c5c04161dd2ff77594ab/html5/thumbnails/12.jpg)
8/19
Cops & Robbers Game [Nowakowski and Winkler; Quilliot, 1983]
Rules of the C&R game
1 Place k ≥ 1 Cops C on nodes
2 Visible Robber R at one node
3 Turn by turn(1) each C slides along ≤ 1 edge(2) R slides along ≤ 1 edge
Goal of the C&R game
Robber must avoid the Cops
Cops must capture Robber (i.e.,occupy the same node)
Cop Number of a graph G
cn(G): min # Cops to win in G
F. Mc Inerney The Game of Cops and Robbers on Graphs
![Page 13: The Game of Cops and Robbers on Graphs - Inria · 2017-11-06 · 1/19 The Game of Cops and Robbers on Graphs Fionn Mc Inerney Universit e C^ote d’Azur, Inria, CNRS, I3S, France](https://reader034.vdocuments.us/reader034/viewer/2022042119/5e97c5c04161dd2ff77594ab/html5/thumbnails/13.jpg)
8/19
Cops & Robbers Game [Nowakowski and Winkler; Quilliot, 1983]
Rules of the C&R game
1 Place k ≥ 1 Cops C on nodes
2 Visible Robber R at one node
3 Turn by turn(1) each C slides along ≤ 1 edge(2) R slides along ≤ 1 edge
Goal of the C&R game
Robber must avoid the Cops
Cops must capture Robber (i.e.,occupy the same node)
Cop Number of a graph G
cn(G): min # Cops to win in G
F. Mc Inerney The Game of Cops and Robbers on Graphs
![Page 14: The Game of Cops and Robbers on Graphs - Inria · 2017-11-06 · 1/19 The Game of Cops and Robbers on Graphs Fionn Mc Inerney Universit e C^ote d’Azur, Inria, CNRS, I3S, France](https://reader034.vdocuments.us/reader034/viewer/2022042119/5e97c5c04161dd2ff77594ab/html5/thumbnails/14.jpg)
8/19
Cops & Robbers Game [Nowakowski and Winkler; Quilliot, 1983]
Rules of the C&R game
1 Place k ≥ 1 Cops C on nodes
2 Visible Robber R at one node
3 Turn by turn(1) each C slides along ≤ 1 edge(2) R slides along ≤ 1 edge
Goal of the C&R game
Robber must avoid the Cops
Cops must capture Robber (i.e.,occupy the same node)
Cop Number of a graph G
cn(G): min # Cops to win in G
F. Mc Inerney The Game of Cops and Robbers on Graphs
![Page 15: The Game of Cops and Robbers on Graphs - Inria · 2017-11-06 · 1/19 The Game of Cops and Robbers on Graphs Fionn Mc Inerney Universit e C^ote d’Azur, Inria, CNRS, I3S, France](https://reader034.vdocuments.us/reader034/viewer/2022042119/5e97c5c04161dd2ff77594ab/html5/thumbnails/15.jpg)
8/19
Cops & Robbers Game [Nowakowski and Winkler; Quilliot, 1983]
Rules of the C&R game
1 Place k ≥ 1 Cops C on nodes
2 Visible Robber R at one node
3 Turn by turn(1) each C slides along ≤ 1 edge
(2) R slides along ≤ 1 edge
Goal of the C&R game
Robber must avoid the Cops
Cops must capture Robber (i.e.,occupy the same node)
Cop Number of a graph G
cn(G): min # Cops to win in G
F. Mc Inerney The Game of Cops and Robbers on Graphs
![Page 16: The Game of Cops and Robbers on Graphs - Inria · 2017-11-06 · 1/19 The Game of Cops and Robbers on Graphs Fionn Mc Inerney Universit e C^ote d’Azur, Inria, CNRS, I3S, France](https://reader034.vdocuments.us/reader034/viewer/2022042119/5e97c5c04161dd2ff77594ab/html5/thumbnails/16.jpg)
8/19
Cops & Robbers Game [Nowakowski and Winkler; Quilliot, 1983]
Rules of the C&R game
1 Place k ≥ 1 Cops C on nodes
2 Visible Robber R at one node
3 Turn by turn(1) each C slides along ≤ 1 edge(2) R slides along ≤ 1 edge
Goal of the C&R game
Robber must avoid the Cops
Cops must capture Robber (i.e.,occupy the same node)
Cop Number of a graph G
cn(G): min # Cops to win in G
F. Mc Inerney The Game of Cops and Robbers on Graphs
![Page 17: The Game of Cops and Robbers on Graphs - Inria · 2017-11-06 · 1/19 The Game of Cops and Robbers on Graphs Fionn Mc Inerney Universit e C^ote d’Azur, Inria, CNRS, I3S, France](https://reader034.vdocuments.us/reader034/viewer/2022042119/5e97c5c04161dd2ff77594ab/html5/thumbnails/17.jpg)
8/19
Cops & Robbers Game [Nowakowski and Winkler; Quilliot, 1983]
Rules of the C&R game
1 Place k ≥ 1 Cops C on nodes
2 Visible Robber R at one node
3 Turn by turn(1) each C slides along ≤ 1 edge(2) R slides along ≤ 1 edge
Goal of the C&R game
Robber must avoid the Cops
Cops must capture Robber (i.e.,occupy the same node)
Cop Number of a graph G
cn(G): min # Cops to win in G
F. Mc Inerney The Game of Cops and Robbers on Graphs
![Page 18: The Game of Cops and Robbers on Graphs - Inria · 2017-11-06 · 1/19 The Game of Cops and Robbers on Graphs Fionn Mc Inerney Universit e C^ote d’Azur, Inria, CNRS, I3S, France](https://reader034.vdocuments.us/reader034/viewer/2022042119/5e97c5c04161dd2ff77594ab/html5/thumbnails/18.jpg)
8/19
Cops & Robbers Game [Nowakowski and Winkler; Quilliot, 1983]
Rules of the C&R game
1 Place k ≥ 1 Cops C on nodes
2 Visible Robber R at one node
3 Turn by turn(1) each C slides along ≤ 1 edge(2) R slides along ≤ 1 edge
Goal of the C&R game
Robber must avoid the Cops
Cops must capture Robber (i.e.,occupy the same node)
Cop Number of a graph G
cn(G): min # Cops to win in G
F. Mc Inerney The Game of Cops and Robbers on Graphs
![Page 19: The Game of Cops and Robbers on Graphs - Inria · 2017-11-06 · 1/19 The Game of Cops and Robbers on Graphs Fionn Mc Inerney Universit e C^ote d’Azur, Inria, CNRS, I3S, France](https://reader034.vdocuments.us/reader034/viewer/2022042119/5e97c5c04161dd2ff77594ab/html5/thumbnails/19.jpg)
8/19
Cops & Robbers Game [Nowakowski and Winkler; Quilliot, 1983]
Rules of the C&R game
1 Place k ≥ 1 Cops C on nodes
2 Visible Robber R at one node
3 Turn by turn(1) each C slides along ≤ 1 edge(2) R slides along ≤ 1 edge
Goal of the C&R game
Robber must avoid the Cops
Cops must capture Robber (i.e.,occupy the same node)
Cop Number of a graph G
cn(G): min # Cops to win in G
F. Mc Inerney The Game of Cops and Robbers on Graphs
![Page 20: The Game of Cops and Robbers on Graphs - Inria · 2017-11-06 · 1/19 The Game of Cops and Robbers on Graphs Fionn Mc Inerney Universit e C^ote d’Azur, Inria, CNRS, I3S, France](https://reader034.vdocuments.us/reader034/viewer/2022042119/5e97c5c04161dd2ff77594ab/html5/thumbnails/20.jpg)
8/19
Cops & Robbers Game [Nowakowski and Winkler; Quilliot, 1983]
Rules of the C&R game
1 Place k ≥ 1 Cops C on nodes
2 Visible Robber R at one node
3 Turn by turn(1) each C slides along ≤ 1 edge(2) R slides along ≤ 1 edge
Goal of the C&R game
Robber must avoid the Cops
Cops must capture Robber (i.e.,occupy the same node)
Cop Number of a graph G
cn(G): min # Cops to win in G
F. Mc Inerney The Game of Cops and Robbers on Graphs
![Page 21: The Game of Cops and Robbers on Graphs - Inria · 2017-11-06 · 1/19 The Game of Cops and Robbers on Graphs Fionn Mc Inerney Universit e C^ote d’Azur, Inria, CNRS, I3S, France](https://reader034.vdocuments.us/reader034/viewer/2022042119/5e97c5c04161dd2ff77594ab/html5/thumbnails/21.jpg)
9/19
Let’s play a bit
Easy remark: For any graph G , cn(G) ≤ γ(G) the size of a min dominating set of G .
F. Mc Inerney The Game of Cops and Robbers on Graphs
![Page 22: The Game of Cops and Robbers on Graphs - Inria · 2017-11-06 · 1/19 The Game of Cops and Robbers on Graphs Fionn Mc Inerney Universit e C^ote d’Azur, Inria, CNRS, I3S, France](https://reader034.vdocuments.us/reader034/viewer/2022042119/5e97c5c04161dd2ff77594ab/html5/thumbnails/22.jpg)
9/19
Let’s play a bit
Easy remark: For any graph G , cn(G) ≤ γ(G) the size of a min dominating set of G .
F. Mc Inerney The Game of Cops and Robbers on Graphs
![Page 23: The Game of Cops and Robbers on Graphs - Inria · 2017-11-06 · 1/19 The Game of Cops and Robbers on Graphs Fionn Mc Inerney Universit e C^ote d’Azur, Inria, CNRS, I3S, France](https://reader034.vdocuments.us/reader034/viewer/2022042119/5e97c5c04161dd2ff77594ab/html5/thumbnails/23.jpg)
9/19
Let’s play a bit
Easy remark: For any graph G , cn(G) ≤ γ(G) the size of a min dominating set of G .
F. Mc Inerney The Game of Cops and Robbers on Graphs
![Page 24: The Game of Cops and Robbers on Graphs - Inria · 2017-11-06 · 1/19 The Game of Cops and Robbers on Graphs Fionn Mc Inerney Universit e C^ote d’Azur, Inria, CNRS, I3S, France](https://reader034.vdocuments.us/reader034/viewer/2022042119/5e97c5c04161dd2ff77594ab/html5/thumbnails/24.jpg)
9/19
Let’s play a bit
Easy remark: For any graph G , cn(G) ≤ γ(G) the size of a min dominating set of G .
F. Mc Inerney The Game of Cops and Robbers on Graphs
![Page 25: The Game of Cops and Robbers on Graphs - Inria · 2017-11-06 · 1/19 The Game of Cops and Robbers on Graphs Fionn Mc Inerney Universit e C^ote d’Azur, Inria, CNRS, I3S, France](https://reader034.vdocuments.us/reader034/viewer/2022042119/5e97c5c04161dd2ff77594ab/html5/thumbnails/25.jpg)
9/19
Let’s play a bit
Easy remark: For any graph G , cn(G) ≤ γ(G) the size of a min dominating set of G .
F. Mc Inerney The Game of Cops and Robbers on Graphs
![Page 26: The Game of Cops and Robbers on Graphs - Inria · 2017-11-06 · 1/19 The Game of Cops and Robbers on Graphs Fionn Mc Inerney Universit e C^ote d’Azur, Inria, CNRS, I3S, France](https://reader034.vdocuments.us/reader034/viewer/2022042119/5e97c5c04161dd2ff77594ab/html5/thumbnails/26.jpg)
9/19
Let’s play a bit
Easy remark: For any graph G , cn(G) ≤ γ(G) the size of a min dominating set of G .
F. Mc Inerney The Game of Cops and Robbers on Graphs
![Page 27: The Game of Cops and Robbers on Graphs - Inria · 2017-11-06 · 1/19 The Game of Cops and Robbers on Graphs Fionn Mc Inerney Universit e C^ote d’Azur, Inria, CNRS, I3S, France](https://reader034.vdocuments.us/reader034/viewer/2022042119/5e97c5c04161dd2ff77594ab/html5/thumbnails/27.jpg)
9/19
Let’s play a bit
cn(tree)=1
Easy remark: For any graph G , cn(G) ≤ γ(G) the size of a min dominating set of G .
F. Mc Inerney The Game of Cops and Robbers on Graphs
![Page 28: The Game of Cops and Robbers on Graphs - Inria · 2017-11-06 · 1/19 The Game of Cops and Robbers on Graphs Fionn Mc Inerney Universit e C^ote d’Azur, Inria, CNRS, I3S, France](https://reader034.vdocuments.us/reader034/viewer/2022042119/5e97c5c04161dd2ff77594ab/html5/thumbnails/28.jpg)
9/19
Let’s play a bit
cn(tree)=1
cn(clique)=?
cn(cycle)=?
cn(Petersen)=?
Easy remark: For any graph G , cn(G) ≤ γ(G) the size of a min dominating set of G .
F. Mc Inerney The Game of Cops and Robbers on Graphs
![Page 29: The Game of Cops and Robbers on Graphs - Inria · 2017-11-06 · 1/19 The Game of Cops and Robbers on Graphs Fionn Mc Inerney Universit e C^ote d’Azur, Inria, CNRS, I3S, France](https://reader034.vdocuments.us/reader034/viewer/2022042119/5e97c5c04161dd2ff77594ab/html5/thumbnails/29.jpg)
9/19
Let’s play a bit
cn(tree)=1
cn(clique)=?
cn(cycle)=?
cn(Petersen)=?
Easy remark: For any graph G , cn(G) ≤ γ(G) the size of a min dominating set of G .
F. Mc Inerney The Game of Cops and Robbers on Graphs
![Page 30: The Game of Cops and Robbers on Graphs - Inria · 2017-11-06 · 1/19 The Game of Cops and Robbers on Graphs Fionn Mc Inerney Universit e C^ote d’Azur, Inria, CNRS, I3S, France](https://reader034.vdocuments.us/reader034/viewer/2022042119/5e97c5c04161dd2ff77594ab/html5/thumbnails/30.jpg)
9/19
Let’s play a bit
cn(tree)=1
cn(clique)=?
cn(cycle)=?
cn(Petersen)=?
Easy remark: For any graph G , cn(G) ≤ γ(G) the size of a min dominating set of G .
F. Mc Inerney The Game of Cops and Robbers on Graphs
![Page 31: The Game of Cops and Robbers on Graphs - Inria · 2017-11-06 · 1/19 The Game of Cops and Robbers on Graphs Fionn Mc Inerney Universit e C^ote d’Azur, Inria, CNRS, I3S, France](https://reader034.vdocuments.us/reader034/viewer/2022042119/5e97c5c04161dd2ff77594ab/html5/thumbnails/31.jpg)
9/19
Let’s play a bit
cn(tree)=1
cn(clique)=1
cn(cycle)=?
cn(Petersen)=?
Easy remark: For any graph G , cn(G) ≤ γ(G) the size of a min dominating set of G .
F. Mc Inerney The Game of Cops and Robbers on Graphs
![Page 32: The Game of Cops and Robbers on Graphs - Inria · 2017-11-06 · 1/19 The Game of Cops and Robbers on Graphs Fionn Mc Inerney Universit e C^ote d’Azur, Inria, CNRS, I3S, France](https://reader034.vdocuments.us/reader034/viewer/2022042119/5e97c5c04161dd2ff77594ab/html5/thumbnails/32.jpg)
9/19
Let’s play a bit
cn(tree)=1
cn(clique)=1
cn(cycle)=?
cn(Petersen)=?
Easy remark: For any graph G , cn(G) ≤ γ(G) the size of a min dominating set of G .
F. Mc Inerney The Game of Cops and Robbers on Graphs
![Page 33: The Game of Cops and Robbers on Graphs - Inria · 2017-11-06 · 1/19 The Game of Cops and Robbers on Graphs Fionn Mc Inerney Universit e C^ote d’Azur, Inria, CNRS, I3S, France](https://reader034.vdocuments.us/reader034/viewer/2022042119/5e97c5c04161dd2ff77594ab/html5/thumbnails/33.jpg)
9/19
Let’s play a bit
cn(tree)=1
cn(clique)=1
cn(cycle)=?
cn(Petersen)=?
Easy remark: For any graph G , cn(G) ≤ γ(G) the size of a min dominating set of G .
F. Mc Inerney The Game of Cops and Robbers on Graphs
![Page 34: The Game of Cops and Robbers on Graphs - Inria · 2017-11-06 · 1/19 The Game of Cops and Robbers on Graphs Fionn Mc Inerney Universit e C^ote d’Azur, Inria, CNRS, I3S, France](https://reader034.vdocuments.us/reader034/viewer/2022042119/5e97c5c04161dd2ff77594ab/html5/thumbnails/34.jpg)
9/19
Let’s play a bit
cn(tree)=1
cn(clique)=1
cn(cycle)=?
cn(Petersen)=?
Easy remark: For any graph G , cn(G) ≤ γ(G) the size of a min dominating set of G .
F. Mc Inerney The Game of Cops and Robbers on Graphs
![Page 35: The Game of Cops and Robbers on Graphs - Inria · 2017-11-06 · 1/19 The Game of Cops and Robbers on Graphs Fionn Mc Inerney Universit e C^ote d’Azur, Inria, CNRS, I3S, France](https://reader034.vdocuments.us/reader034/viewer/2022042119/5e97c5c04161dd2ff77594ab/html5/thumbnails/35.jpg)
9/19
Let’s play a bit
cn(tree)=1
cn(clique)=1
cn(cycle)=?
cn(Petersen)=?
Easy remark: For any graph G , cn(G) ≤ γ(G) the size of a min dominating set of G .
F. Mc Inerney The Game of Cops and Robbers on Graphs
![Page 36: The Game of Cops and Robbers on Graphs - Inria · 2017-11-06 · 1/19 The Game of Cops and Robbers on Graphs Fionn Mc Inerney Universit e C^ote d’Azur, Inria, CNRS, I3S, France](https://reader034.vdocuments.us/reader034/viewer/2022042119/5e97c5c04161dd2ff77594ab/html5/thumbnails/36.jpg)
9/19
Let’s play a bit
cn(tree)=1
cn(clique)=1
cn(cycle)=?
cn(Petersen)=?
Easy remark: For any graph G , cn(G) ≤ γ(G) the size of a min dominating set of G .
F. Mc Inerney The Game of Cops and Robbers on Graphs
![Page 37: The Game of Cops and Robbers on Graphs - Inria · 2017-11-06 · 1/19 The Game of Cops and Robbers on Graphs Fionn Mc Inerney Universit e C^ote d’Azur, Inria, CNRS, I3S, France](https://reader034.vdocuments.us/reader034/viewer/2022042119/5e97c5c04161dd2ff77594ab/html5/thumbnails/37.jpg)
9/19
Let’s play a bit
cn(tree)=1
cn(clique)=1
cn(cycle)=?
cn(Petersen)=?
Easy remark: For any graph G , cn(G) ≤ γ(G) the size of a min dominating set of G .
F. Mc Inerney The Game of Cops and Robbers on Graphs
![Page 38: The Game of Cops and Robbers on Graphs - Inria · 2017-11-06 · 1/19 The Game of Cops and Robbers on Graphs Fionn Mc Inerney Universit e C^ote d’Azur, Inria, CNRS, I3S, France](https://reader034.vdocuments.us/reader034/viewer/2022042119/5e97c5c04161dd2ff77594ab/html5/thumbnails/38.jpg)
9/19
Let’s play a bit
cn(tree)=1
cn(clique)=1
cn(cycle)=2
cn(Petersen)=?
Easy remark: For any graph G , cn(G) ≤ γ(G) the size of a min dominating set of G .
F. Mc Inerney The Game of Cops and Robbers on Graphs
![Page 39: The Game of Cops and Robbers on Graphs - Inria · 2017-11-06 · 1/19 The Game of Cops and Robbers on Graphs Fionn Mc Inerney Universit e C^ote d’Azur, Inria, CNRS, I3S, France](https://reader034.vdocuments.us/reader034/viewer/2022042119/5e97c5c04161dd2ff77594ab/html5/thumbnails/39.jpg)
9/19
Let’s play a bit
cn(tree)=1
cn(clique)=1
cn(cycle)=2
cn(Petersen)=?
Easy remark: For any graph G , cn(G) ≤ γ(G) the size of a min dominating set of G .
F. Mc Inerney The Game of Cops and Robbers on Graphs
![Page 40: The Game of Cops and Robbers on Graphs - Inria · 2017-11-06 · 1/19 The Game of Cops and Robbers on Graphs Fionn Mc Inerney Universit e C^ote d’Azur, Inria, CNRS, I3S, France](https://reader034.vdocuments.us/reader034/viewer/2022042119/5e97c5c04161dd2ff77594ab/html5/thumbnails/40.jpg)
9/19
Let’s play a bit
cn(tree)=1
cn(clique)=1
cn(cycle)=2
cn(Petersen)=?
Easy remark: For any graph G , cn(G) ≤ γ(G) the size of a min dominating set of G .
F. Mc Inerney The Game of Cops and Robbers on Graphs
![Page 41: The Game of Cops and Robbers on Graphs - Inria · 2017-11-06 · 1/19 The Game of Cops and Robbers on Graphs Fionn Mc Inerney Universit e C^ote d’Azur, Inria, CNRS, I3S, France](https://reader034.vdocuments.us/reader034/viewer/2022042119/5e97c5c04161dd2ff77594ab/html5/thumbnails/41.jpg)
9/19
Let’s play a bit
cn(tree)=1
cn(clique)=1
cn(cycle)=2
cn(Petersen)=?
Easy remark: For any graph G , cn(G) ≤ γ(G) the size of a min dominating set of G .
F. Mc Inerney The Game of Cops and Robbers on Graphs
![Page 42: The Game of Cops and Robbers on Graphs - Inria · 2017-11-06 · 1/19 The Game of Cops and Robbers on Graphs Fionn Mc Inerney Universit e C^ote d’Azur, Inria, CNRS, I3S, France](https://reader034.vdocuments.us/reader034/viewer/2022042119/5e97c5c04161dd2ff77594ab/html5/thumbnails/42.jpg)
9/19
Let’s play a bit
cn(tree)=1
cn(clique)=1
cn(cycle)=2
cn(Petersen)=?
Easy remark: For any graph G , cn(G) ≤ γ(G) the size of a min dominating set of G .
F. Mc Inerney The Game of Cops and Robbers on Graphs
![Page 43: The Game of Cops and Robbers on Graphs - Inria · 2017-11-06 · 1/19 The Game of Cops and Robbers on Graphs Fionn Mc Inerney Universit e C^ote d’Azur, Inria, CNRS, I3S, France](https://reader034.vdocuments.us/reader034/viewer/2022042119/5e97c5c04161dd2ff77594ab/html5/thumbnails/43.jpg)
9/19
Let’s play a bit
cn(tree)=1
cn(clique)=1
cn(cycle)=2
cn(Petersen)=?
Easy remark: For any graph G , cn(G) ≤ γ(G) the size of a min dominating set of G .
F. Mc Inerney The Game of Cops and Robbers on Graphs
![Page 44: The Game of Cops and Robbers on Graphs - Inria · 2017-11-06 · 1/19 The Game of Cops and Robbers on Graphs Fionn Mc Inerney Universit e C^ote d’Azur, Inria, CNRS, I3S, France](https://reader034.vdocuments.us/reader034/viewer/2022042119/5e97c5c04161dd2ff77594ab/html5/thumbnails/44.jpg)
9/19
Let’s play a bit
cn(tree)=1
cn(clique)=1
cn(cycle)=2
cn(Petersen)=3
Easy remark: For any graph G , cn(G) ≤ γ(G) the size of a min dominating set of G .
F. Mc Inerney The Game of Cops and Robbers on Graphs
![Page 45: The Game of Cops and Robbers on Graphs - Inria · 2017-11-06 · 1/19 The Game of Cops and Robbers on Graphs Fionn Mc Inerney Universit e C^ote d’Azur, Inria, CNRS, I3S, France](https://reader034.vdocuments.us/reader034/viewer/2022042119/5e97c5c04161dd2ff77594ab/html5/thumbnails/45.jpg)
10/19
Complexity: a graph G , cn(G ) ≤ k?
Seminal paper: k = 1 [Nowakowski and Winkler; Quilliot, 1983]
cn(G) = 1 iff V = v1, · · · , vn and, ∀i < n, ∃j > i s.t., N[vi ] ⊆ N[vj ].
(dismantable graphs) can be checked in time O(n3)
Generalization to any k [Berarducci, Intrigila’93] [Hahn, MacGillivray’06] [Clarke, MacGillivray’12]
cn(G) ≤ k? can be checked in time nO(k) ∈ EXPTIME
EXPTIME-complete in directed graphs [Goldstein and Reingold, 1995]
NP-hard and W[2]-hard [Fomin,Golovach,Kratochvil,Nisse,Suchan, 2010]
(i.e., no algorithm in time f (k)nO(1) expected)
PSPACE-hard [Mamino 2013]
EXPTIME-complete [Kinnersley 2014]
F. Mc Inerney The Game of Cops and Robbers on Graphs
![Page 46: The Game of Cops and Robbers on Graphs - Inria · 2017-11-06 · 1/19 The Game of Cops and Robbers on Graphs Fionn Mc Inerney Universit e C^ote d’Azur, Inria, CNRS, I3S, France](https://reader034.vdocuments.us/reader034/viewer/2022042119/5e97c5c04161dd2ff77594ab/html5/thumbnails/46.jpg)
10/19
Complexity: a graph G , cn(G ) ≤ k?
Seminal paper: k = 1 [Nowakowski and Winkler; Quilliot, 1983]
cn(G) = 1 iff V = v1, · · · , vn and, ∀i < n, ∃j > i s.t., N[vi ] ⊆ N[vj ].
(dismantable graphs) can be checked in time O(n3)
Generalization to any k [Berarducci, Intrigila’93] [Hahn, MacGillivray’06] [Clarke, MacGillivray’12]
cn(G) ≤ k? can be checked in time nO(k) ∈ EXPTIME
EXPTIME-complete in directed graphs [Goldstein and Reingold, 1995]
NP-hard and W[2]-hard [Fomin,Golovach,Kratochvil,Nisse,Suchan, 2010]
(i.e., no algorithm in time f (k)nO(1) expected)
PSPACE-hard [Mamino 2013]
EXPTIME-complete [Kinnersley 2014]
F. Mc Inerney The Game of Cops and Robbers on Graphs
![Page 47: The Game of Cops and Robbers on Graphs - Inria · 2017-11-06 · 1/19 The Game of Cops and Robbers on Graphs Fionn Mc Inerney Universit e C^ote d’Azur, Inria, CNRS, I3S, France](https://reader034.vdocuments.us/reader034/viewer/2022042119/5e97c5c04161dd2ff77594ab/html5/thumbnails/47.jpg)
10/19
Complexity: a graph G , cn(G ) ≤ k?
Seminal paper: k = 1 [Nowakowski and Winkler; Quilliot, 1983]
cn(G) = 1 iff V = v1, · · · , vn and, ∀i < n, ∃j > i s.t., N[vi ] ⊆ N[vj ].
(dismantable graphs) can be checked in time O(n3)
Generalization to any k [Berarducci, Intrigila’93] [Hahn, MacGillivray’06] [Clarke, MacGillivray’12]
cn(G) ≤ k? can be checked in time nO(k) ∈ EXPTIME
EXPTIME-complete in directed graphs [Goldstein and Reingold, 1995]
NP-hard and W[2]-hard [Fomin,Golovach,Kratochvil,Nisse,Suchan, 2010]
(i.e., no algorithm in time f (k)nO(1) expected)
PSPACE-hard [Mamino 2013]
EXPTIME-complete [Kinnersley 2014]
F. Mc Inerney The Game of Cops and Robbers on Graphs
![Page 48: The Game of Cops and Robbers on Graphs - Inria · 2017-11-06 · 1/19 The Game of Cops and Robbers on Graphs Fionn Mc Inerney Universit e C^ote d’Azur, Inria, CNRS, I3S, France](https://reader034.vdocuments.us/reader034/viewer/2022042119/5e97c5c04161dd2ff77594ab/html5/thumbnails/48.jpg)
10/19
Complexity: a graph G , cn(G ) ≤ k?
Seminal paper: k = 1 [Nowakowski and Winkler; Quilliot, 1983]
cn(G) = 1 iff V = v1, · · · , vn and, ∀i < n, ∃j > i s.t., N[vi ] ⊆ N[vj ].
(dismantable graphs) can be checked in time O(n3)
Generalization to any k [Berarducci, Intrigila’93] [Hahn, MacGillivray’06] [Clarke, MacGillivray’12]
cn(G) ≤ k? can be checked in time nO(k) ∈ EXPTIME
EXPTIME-complete in directed graphs [Goldstein and Reingold, 1995]
NP-hard and W[2]-hard [Fomin,Golovach,Kratochvil,Nisse,Suchan, 2010]
(i.e., no algorithm in time f (k)nO(1) expected)
PSPACE-hard [Mamino 2013]
EXPTIME-complete [Kinnersley 2014]
F. Mc Inerney The Game of Cops and Robbers on Graphs
![Page 49: The Game of Cops and Robbers on Graphs - Inria · 2017-11-06 · 1/19 The Game of Cops and Robbers on Graphs Fionn Mc Inerney Universit e C^ote d’Azur, Inria, CNRS, I3S, France](https://reader034.vdocuments.us/reader034/viewer/2022042119/5e97c5c04161dd2ff77594ab/html5/thumbnails/49.jpg)
10/19
Complexity: a graph G , cn(G ) ≤ k?
Seminal paper: k = 1 [Nowakowski and Winkler; Quilliot, 1983]
cn(G) = 1 iff V = v1, · · · , vn and, ∀i < n, ∃j > i s.t., N[vi ] ⊆ N[vj ].
(dismantable graphs) can be checked in time O(n3)
Generalization to any k [Berarducci, Intrigila’93] [Hahn, MacGillivray’06] [Clarke, MacGillivray’12]
cn(G) ≤ k? can be checked in time nO(k) ∈ EXPTIME
EXPTIME-complete in directed graphs [Goldstein and Reingold, 1995]
NP-hard and W[2]-hard [Fomin,Golovach,Kratochvil,Nisse,Suchan, 2010]
(i.e., no algorithm in time f (k)nO(1) expected)
PSPACE-hard [Mamino 2013]
EXPTIME-complete [Kinnersley 2014]
F. Mc Inerney The Game of Cops and Robbers on Graphs
![Page 50: The Game of Cops and Robbers on Graphs - Inria · 2017-11-06 · 1/19 The Game of Cops and Robbers on Graphs Fionn Mc Inerney Universit e C^ote d’Azur, Inria, CNRS, I3S, France](https://reader034.vdocuments.us/reader034/viewer/2022042119/5e97c5c04161dd2ff77594ab/html5/thumbnails/50.jpg)
10/19
Complexity: a graph G , cn(G ) ≤ k?
Seminal paper: k = 1 [Nowakowski and Winkler; Quilliot, 1983]
cn(G) = 1 iff V = v1, · · · , vn and, ∀i < n, ∃j > i s.t., N[vi ] ⊆ N[vj ].
(dismantable graphs) can be checked in time O(n3)
Generalization to any k [Berarducci, Intrigila’93] [Hahn, MacGillivray’06] [Clarke, MacGillivray’12]
cn(G) ≤ k? can be checked in time nO(k) ∈ EXPTIME
EXPTIME-complete in directed graphs [Goldstein and Reingold, 1995]
NP-hard and W[2]-hard [Fomin,Golovach,Kratochvil,Nisse,Suchan, 2010]
(i.e., no algorithm in time f (k)nO(1) expected)
PSPACE-hard [Mamino 2013]
EXPTIME-complete [Kinnersley 2014]
F. Mc Inerney The Game of Cops and Robbers on Graphs
![Page 51: The Game of Cops and Robbers on Graphs - Inria · 2017-11-06 · 1/19 The Game of Cops and Robbers on Graphs Fionn Mc Inerney Universit e C^ote d’Azur, Inria, CNRS, I3S, France](https://reader034.vdocuments.us/reader034/viewer/2022042119/5e97c5c04161dd2ff77594ab/html5/thumbnails/51.jpg)
11/19
Graphs with high cop-number
Large girth (smallest cycle) AND large min degree ⇒ large cop-number
G with min-degree d and girth > 4 ⇒ cn(G) ≥ d . [Aigner and Fromme 84]
for any d , there are d-regular graphs G with cn(G) ≥ d [Aigner and Fromme 84]
cn(G) ≥ d t in any graph with min-degree d and girth > 8t − 3 [Frankl 87]
for any k, there is G with diameter 2 and cn(G) ≥ k (e.g., Kneser graph KG3k,k )
F. Mc Inerney The Game of Cops and Robbers on Graphs
![Page 52: The Game of Cops and Robbers on Graphs - Inria · 2017-11-06 · 1/19 The Game of Cops and Robbers on Graphs Fionn Mc Inerney Universit e C^ote d’Azur, Inria, CNRS, I3S, France](https://reader034.vdocuments.us/reader034/viewer/2022042119/5e97c5c04161dd2ff77594ab/html5/thumbnails/52.jpg)
11/19
Graphs with high cop-number
Large girth (smallest cycle) AND large min degree ⇒ large cop-number
G with min-degree d and girth > 4 ⇒ cn(G) ≥ d . [Aigner and Fromme 84]
for any d , there are d-regular graphs G with cn(G) ≥ d [Aigner and Fromme 84]
cn(G) ≥ d t in any graph with min-degree d and girth > 8t − 3 [Frankl 87]
for any k, there is G with diameter 2 and cn(G) ≥ k (e.g., Kneser graph KG3k,k )
F. Mc Inerney The Game of Cops and Robbers on Graphs
![Page 53: The Game of Cops and Robbers on Graphs - Inria · 2017-11-06 · 1/19 The Game of Cops and Robbers on Graphs Fionn Mc Inerney Universit e C^ote d’Azur, Inria, CNRS, I3S, France](https://reader034.vdocuments.us/reader034/viewer/2022042119/5e97c5c04161dd2ff77594ab/html5/thumbnails/53.jpg)
12/19
Meyniel’s Conjecture
∃ n-node graphs with min degree Θ(√n) and girth > 4
⇒ ∃ n-node graphs G with cn(G) = Ω(√n)
(e.g., projective plane, random√n-regular graphs)
Meyniel’s Conjecture
Conjecture: For any n-node connected graph G , cn(G) = O(√n) [Meyniel 85]
F. Mc Inerney The Game of Cops and Robbers on Graphs
![Page 54: The Game of Cops and Robbers on Graphs - Inria · 2017-11-06 · 1/19 The Game of Cops and Robbers on Graphs Fionn Mc Inerney Universit e C^ote d’Azur, Inria, CNRS, I3S, France](https://reader034.vdocuments.us/reader034/viewer/2022042119/5e97c5c04161dd2ff77594ab/html5/thumbnails/54.jpg)
13/19
Link with Graph Structural Properties
Reminder: For any graph G , cn(G) ≤ γ(G) the dominating number of G .
Lemma [Aigner, Fromme 1984]
1 Cop is sufficient to “protect” a shortest path P in any graph.(after a finite number of steps, Robber cannot reach P)
⇒ cn(grid) = 2 (while γ(grid) ≈ n/5)
⇒ Cop-number related to structural properties
F. Mc Inerney The Game of Cops and Robbers on Graphs
![Page 55: The Game of Cops and Robbers on Graphs - Inria · 2017-11-06 · 1/19 The Game of Cops and Robbers on Graphs Fionn Mc Inerney Universit e C^ote d’Azur, Inria, CNRS, I3S, France](https://reader034.vdocuments.us/reader034/viewer/2022042119/5e97c5c04161dd2ff77594ab/html5/thumbnails/55.jpg)
13/19
Link with Graph Structural Properties
Reminder: For any graph G , cn(G) ≤ γ(G) the dominating number of G .
Lemma [Aigner, Fromme 1984]
1 Cop is sufficient to “protect” a shortest path P in any graph.(after a finite number of steps, Robber cannot reach P)
⇒ cn(grid) = 2 (while γ(grid) ≈ n/5)
⇒ Cop-number related to structural properties
F. Mc Inerney The Game of Cops and Robbers on Graphs
![Page 56: The Game of Cops and Robbers on Graphs - Inria · 2017-11-06 · 1/19 The Game of Cops and Robbers on Graphs Fionn Mc Inerney Universit e C^ote d’Azur, Inria, CNRS, I3S, France](https://reader034.vdocuments.us/reader034/viewer/2022042119/5e97c5c04161dd2ff77594ab/html5/thumbnails/56.jpg)
13/19
Link with Graph Structural Properties
Reminder: For any graph G , cn(G) ≤ γ(G) the dominating number of G .
Lemma [Aigner, Fromme 1984]
1 Cop is sufficient to “protect” a shortest path P in any graph.(after a finite number of steps, Robber cannot reach P)
⇒ cn(grid) = 2 (while γ(grid) ≈ n/5)
⇒ Cop-number related to structural properties
F. Mc Inerney The Game of Cops and Robbers on Graphs
![Page 57: The Game of Cops and Robbers on Graphs - Inria · 2017-11-06 · 1/19 The Game of Cops and Robbers on Graphs Fionn Mc Inerney Universit e C^ote d’Azur, Inria, CNRS, I3S, France](https://reader034.vdocuments.us/reader034/viewer/2022042119/5e97c5c04161dd2ff77594ab/html5/thumbnails/57.jpg)
13/19
Link with Graph Structural Properties
Reminder: For any graph G , cn(G) ≤ γ(G) the dominating number of G .
Lemma [Aigner, Fromme 1984]
1 Cop is sufficient to “protect” a shortest path P in any graph.(after a finite number of steps, Robber cannot reach P)
⇒ cn(grid) = 2 (while γ(grid) ≈ n/5)
⇒ Cop-number related to structural properties
F. Mc Inerney The Game of Cops and Robbers on Graphs
![Page 58: The Game of Cops and Robbers on Graphs - Inria · 2017-11-06 · 1/19 The Game of Cops and Robbers on Graphs Fionn Mc Inerney Universit e C^ote d’Azur, Inria, CNRS, I3S, France](https://reader034.vdocuments.us/reader034/viewer/2022042119/5e97c5c04161dd2ff77594ab/html5/thumbnails/58.jpg)
13/19
Link with Graph Structural Properties
Reminder: For any graph G , cn(G) ≤ γ(G) the dominating number of G .
Lemma [Aigner, Fromme 1984]
1 Cop is sufficient to “protect” a shortest path P in any graph.(after a finite number of steps, Robber cannot reach P)
⇒ cn(grid) = 2 (while γ(grid) ≈ n/5)
⇒ Cop-number related to structural properties
F. Mc Inerney The Game of Cops and Robbers on Graphs
![Page 59: The Game of Cops and Robbers on Graphs - Inria · 2017-11-06 · 1/19 The Game of Cops and Robbers on Graphs Fionn Mc Inerney Universit e C^ote d’Azur, Inria, CNRS, I3S, France](https://reader034.vdocuments.us/reader034/viewer/2022042119/5e97c5c04161dd2ff77594ab/html5/thumbnails/59.jpg)
13/19
Link with Graph Structural Properties
Reminder: For any graph G , cn(G) ≤ γ(G) the dominating number of G .
Lemma [Aigner, Fromme 1984]
1 Cop is sufficient to “protect” a shortest path P in any graph.(after a finite number of steps, Robber cannot reach P)
⇒ cn(grid) = 2 (while γ(grid) ≈ n/5)
⇒ Cop-number related to structural properties
F. Mc Inerney The Game of Cops and Robbers on Graphs
![Page 60: The Game of Cops and Robbers on Graphs - Inria · 2017-11-06 · 1/19 The Game of Cops and Robbers on Graphs Fionn Mc Inerney Universit e C^ote d’Azur, Inria, CNRS, I3S, France](https://reader034.vdocuments.us/reader034/viewer/2022042119/5e97c5c04161dd2ff77594ab/html5/thumbnails/60.jpg)
13/19
Link with Graph Structural Properties
Reminder: For any graph G , cn(G) ≤ γ(G) the dominating number of G .
Lemma [Aigner, Fromme 1984]
1 Cop is sufficient to “protect” a shortest path P in any graph.(after a finite number of steps, Robber cannot reach P)
⇒ cn(grid) = 2 (while γ(grid) ≈ n/5)
⇒ Cop-number related to structural properties
F. Mc Inerney The Game of Cops and Robbers on Graphs
![Page 61: The Game of Cops and Robbers on Graphs - Inria · 2017-11-06 · 1/19 The Game of Cops and Robbers on Graphs Fionn Mc Inerney Universit e C^ote d’Azur, Inria, CNRS, I3S, France](https://reader034.vdocuments.us/reader034/viewer/2022042119/5e97c5c04161dd2ff77594ab/html5/thumbnails/61.jpg)
13/19
Link with Graph Structural Properties
Reminder: For any graph G , cn(G) ≤ γ(G) the dominating number of G .
Lemma [Aigner, Fromme 1984]
1 Cop is sufficient to “protect” a shortest path P in any graph.(after a finite number of steps, Robber cannot reach P)
⇒ cn(grid) = 2 (while γ(grid) ≈ n/5)
⇒ Cop-number related to structural properties
F. Mc Inerney The Game of Cops and Robbers on Graphs
![Page 62: The Game of Cops and Robbers on Graphs - Inria · 2017-11-06 · 1/19 The Game of Cops and Robbers on Graphs Fionn Mc Inerney Universit e C^ote d’Azur, Inria, CNRS, I3S, France](https://reader034.vdocuments.us/reader034/viewer/2022042119/5e97c5c04161dd2ff77594ab/html5/thumbnails/62.jpg)
13/19
Link with Graph Structural Properties
Reminder: For any graph G , cn(G) ≤ γ(G) the dominating number of G .
Lemma [Aigner, Fromme 1984]
1 Cop is sufficient to “protect” a shortest path P in any graph.(after a finite number of steps, Robber cannot reach P)
⇒ cn(grid) = 2 (while γ(grid) ≈ n/5)
⇒ Cop-number related to structural properties
F. Mc Inerney The Game of Cops and Robbers on Graphs
![Page 63: The Game of Cops and Robbers on Graphs - Inria · 2017-11-06 · 1/19 The Game of Cops and Robbers on Graphs Fionn Mc Inerney Universit e C^ote d’Azur, Inria, CNRS, I3S, France](https://reader034.vdocuments.us/reader034/viewer/2022042119/5e97c5c04161dd2ff77594ab/html5/thumbnails/63.jpg)
13/19
Link with Graph Structural Properties
Reminder: For any graph G , cn(G) ≤ γ(G) the dominating number of G .
Lemma [Aigner, Fromme 1984]
1 Cop is sufficient to “protect” a shortest path P in any graph.(after a finite number of steps, Robber cannot reach P)
⇒ cn(grid) = 2 (while γ(grid) ≈ n/5)
⇒ Cop-number related to structural properties
F. Mc Inerney The Game of Cops and Robbers on Graphs
![Page 64: The Game of Cops and Robbers on Graphs - Inria · 2017-11-06 · 1/19 The Game of Cops and Robbers on Graphs Fionn Mc Inerney Universit e C^ote d’Azur, Inria, CNRS, I3S, France](https://reader034.vdocuments.us/reader034/viewer/2022042119/5e97c5c04161dd2ff77594ab/html5/thumbnails/64.jpg)
13/19
Link with Graph Structural Properties
Reminder: For any graph G , cn(G) ≤ γ(G) the dominating number of G .
Lemma [Aigner, Fromme 1984]
1 Cop is sufficient to “protect” a shortest path P in any graph.(after a finite number of steps, Robber cannot reach P)
⇒ cn(grid) = 2 (while γ(grid) ≈ n/5)
⇒ Cop-number related to structural properties
F. Mc Inerney The Game of Cops and Robbers on Graphs
![Page 65: The Game of Cops and Robbers on Graphs - Inria · 2017-11-06 · 1/19 The Game of Cops and Robbers on Graphs Fionn Mc Inerney Universit e C^ote d’Azur, Inria, CNRS, I3S, France](https://reader034.vdocuments.us/reader034/viewer/2022042119/5e97c5c04161dd2ff77594ab/html5/thumbnails/65.jpg)
13/19
Link with Graph Structural Properties
Reminder: For any graph G , cn(G) ≤ γ(G) the dominating number of G .
Lemma [Aigner, Fromme 1984]
1 Cop is sufficient to “protect” a shortest path P in any graph.(after a finite number of steps, Robber cannot reach P)
⇒ cn(grid) = 2 (while γ(grid) ≈ n/5)
⇒ Cop-number related to structural properties
F. Mc Inerney The Game of Cops and Robbers on Graphs
![Page 66: The Game of Cops and Robbers on Graphs - Inria · 2017-11-06 · 1/19 The Game of Cops and Robbers on Graphs Fionn Mc Inerney Universit e C^ote d’Azur, Inria, CNRS, I3S, France](https://reader034.vdocuments.us/reader034/viewer/2022042119/5e97c5c04161dd2ff77594ab/html5/thumbnails/66.jpg)
13/19
Link with Graph Structural Properties
Reminder: For any graph G , cn(G) ≤ γ(G) the dominating number of G .
Lemma [Aigner, Fromme 1984]
1 Cop is sufficient to “protect” a shortest path P in any graph.(after a finite number of steps, Robber cannot reach P)
⇒ cn(grid) = 2 (while γ(grid) ≈ n/5)
⇒ Cop-number related to structural properties
F. Mc Inerney The Game of Cops and Robbers on Graphs
![Page 67: The Game of Cops and Robbers on Graphs - Inria · 2017-11-06 · 1/19 The Game of Cops and Robbers on Graphs Fionn Mc Inerney Universit e C^ote d’Azur, Inria, CNRS, I3S, France](https://reader034.vdocuments.us/reader034/viewer/2022042119/5e97c5c04161dd2ff77594ab/html5/thumbnails/67.jpg)
13/19
Link with Graph Structural Properties
Reminder: For any graph G , cn(G) ≤ γ(G) the dominating number of G .
Lemma [Aigner, Fromme 1984]
1 Cop is sufficient to “protect” a shortest path P in any graph.(after a finite number of steps, Robber cannot reach P)
⇒ cn(grid) = 2 (while γ(grid) ≈ n/5)
⇒ Cop-number related to structural properties
F. Mc Inerney The Game of Cops and Robbers on Graphs
![Page 68: The Game of Cops and Robbers on Graphs - Inria · 2017-11-06 · 1/19 The Game of Cops and Robbers on Graphs Fionn Mc Inerney Universit e C^ote d’Azur, Inria, CNRS, I3S, France](https://reader034.vdocuments.us/reader034/viewer/2022042119/5e97c5c04161dd2ff77594ab/html5/thumbnails/68.jpg)
13/19
Link with Graph Structural Properties
Reminder: For any graph G , cn(G) ≤ γ(G) the dominating number of G .
Lemma [Aigner, Fromme 1984]
1 Cop is sufficient to “protect” a shortest path P in any graph.(after a finite number of steps, Robber cannot reach P)
⇒ cn(grid) = 2 (while γ(grid) ≈ n/5)
⇒ Cop-number related to structural properties
F. Mc Inerney The Game of Cops and Robbers on Graphs
![Page 69: The Game of Cops and Robbers on Graphs - Inria · 2017-11-06 · 1/19 The Game of Cops and Robbers on Graphs Fionn Mc Inerney Universit e C^ote d’Azur, Inria, CNRS, I3S, France](https://reader034.vdocuments.us/reader034/viewer/2022042119/5e97c5c04161dd2ff77594ab/html5/thumbnails/69.jpg)
13/19
Link with Graph Structural Properties
Reminder: For any graph G , cn(G) ≤ γ(G) the dominating number of G .
Lemma [Aigner, Fromme 1984]
1 Cop is sufficient to “protect” a shortest path P in any graph.(after a finite number of steps, Robber cannot reach P)
⇒ cn(grid) = 2 (while γ(grid) ≈ n/5)
⇒ Cop-number related to structural properties
F. Mc Inerney The Game of Cops and Robbers on Graphs
![Page 70: The Game of Cops and Robbers on Graphs - Inria · 2017-11-06 · 1/19 The Game of Cops and Robbers on Graphs Fionn Mc Inerney Universit e C^ote d’Azur, Inria, CNRS, I3S, France](https://reader034.vdocuments.us/reader034/viewer/2022042119/5e97c5c04161dd2ff77594ab/html5/thumbnails/70.jpg)
13/19
Link with Graph Structural Properties
Reminder: For any graph G , cn(G) ≤ γ(G) the dominating number of G .
Lemma [Aigner, Fromme 1984]
1 Cop is sufficient to “protect” a shortest path P in any graph.(after a finite number of steps, Robber cannot reach P)
⇒ cn(grid) = 2 (while γ(grid) ≈ n/5)
⇒ Cop-number related to structural properties
F. Mc Inerney The Game of Cops and Robbers on Graphs
![Page 71: The Game of Cops and Robbers on Graphs - Inria · 2017-11-06 · 1/19 The Game of Cops and Robbers on Graphs Fionn Mc Inerney Universit e C^ote d’Azur, Inria, CNRS, I3S, France](https://reader034.vdocuments.us/reader034/viewer/2022042119/5e97c5c04161dd2ff77594ab/html5/thumbnails/71.jpg)
13/19
Link with Graph Structural Properties
Reminder: For any graph G , cn(G) ≤ γ(G) the dominating number of G .
Lemma [Aigner, Fromme 1984]
1 Cop is sufficient to “protect” a shortest path P in any graph.(after a finite number of steps, Robber cannot reach P)
⇒ cn(grid) = 2 (while γ(grid) ≈ n/5)
⇒ Cop-number related to structural properties
F. Mc Inerney The Game of Cops and Robbers on Graphs
![Page 72: The Game of Cops and Robbers on Graphs - Inria · 2017-11-06 · 1/19 The Game of Cops and Robbers on Graphs Fionn Mc Inerney Universit e C^ote d’Azur, Inria, CNRS, I3S, France](https://reader034.vdocuments.us/reader034/viewer/2022042119/5e97c5c04161dd2ff77594ab/html5/thumbnails/72.jpg)
13/19
Link with Graph Structural Properties
Reminder: For any graph G , cn(G) ≤ γ(G) the dominating number of G .
Lemma [Aigner, Fromme 1984]
1 Cop is sufficient to “protect” a shortest path P in any graph.(after a finite number of steps, Robber cannot reach P)
⇒ cn(grid) = 2 (while γ(grid) ≈ n/5)
⇒ Cop-number related to structural properties
F. Mc Inerney The Game of Cops and Robbers on Graphs
![Page 73: The Game of Cops and Robbers on Graphs - Inria · 2017-11-06 · 1/19 The Game of Cops and Robbers on Graphs Fionn Mc Inerney Universit e C^ote d’Azur, Inria, CNRS, I3S, France](https://reader034.vdocuments.us/reader034/viewer/2022042119/5e97c5c04161dd2ff77594ab/html5/thumbnails/73.jpg)
13/19
Link with Graph Structural Properties
Reminder: For any graph G , cn(G) ≤ γ(G) the dominating number of G .
Lemma [Aigner, Fromme 1984]
1 Cop is sufficient to “protect” a shortest path P in any graph.(after a finite number of steps, Robber cannot reach P)
⇒ cn(grid) = 2 (while γ(grid) ≈ n/5)
⇒ Cop-number related to structural properties
F. Mc Inerney The Game of Cops and Robbers on Graphs
![Page 74: The Game of Cops and Robbers on Graphs - Inria · 2017-11-06 · 1/19 The Game of Cops and Robbers on Graphs Fionn Mc Inerney Universit e C^ote d’Azur, Inria, CNRS, I3S, France](https://reader034.vdocuments.us/reader034/viewer/2022042119/5e97c5c04161dd2ff77594ab/html5/thumbnails/74.jpg)
13/19
Link with Graph Structural Properties
Reminder: For any graph G , cn(G) ≤ γ(G) the dominating number of G .
Lemma [Aigner, Fromme 1984]
1 Cop is sufficient to “protect” a shortest path P in any graph.(after a finite number of steps, Robber cannot reach P)
⇒ cn(grid) = 2 (while γ(grid) ≈ n/5)
⇒ Cop-number related to structural properties
F. Mc Inerney The Game of Cops and Robbers on Graphs
![Page 75: The Game of Cops and Robbers on Graphs - Inria · 2017-11-06 · 1/19 The Game of Cops and Robbers on Graphs Fionn Mc Inerney Universit e C^ote d’Azur, Inria, CNRS, I3S, France](https://reader034.vdocuments.us/reader034/viewer/2022042119/5e97c5c04161dd2ff77594ab/html5/thumbnails/75.jpg)
13/19
Link with Graph Structural Properties
Reminder: For any graph G , cn(G) ≤ γ(G) the dominating number of G .
Lemma [Aigner, Fromme 1984]
1 Cop is sufficient to “protect” a shortest path P in any graph.(after a finite number of steps, Robber cannot reach P)
⇒ cn(grid) = 2 (while γ(grid) ≈ n/5)
⇒ Cop-number related to structural properties
F. Mc Inerney The Game of Cops and Robbers on Graphs
![Page 76: The Game of Cops and Robbers on Graphs - Inria · 2017-11-06 · 1/19 The Game of Cops and Robbers on Graphs Fionn Mc Inerney Universit e C^ote d’Azur, Inria, CNRS, I3S, France](https://reader034.vdocuments.us/reader034/viewer/2022042119/5e97c5c04161dd2ff77594ab/html5/thumbnails/76.jpg)
13/19
Link with Graph Structural Properties
Reminder: For any graph G , cn(G) ≤ γ(G) the dominating number of G .
Lemma [Aigner, Fromme 1984]
1 Cop is sufficient to “protect” a shortest path P in any graph.(after a finite number of steps, Robber cannot reach P)
⇒ cn(grid) = 2 (while γ(grid) ≈ n/5)
⇒ Cop-number related to structural properties
F. Mc Inerney The Game of Cops and Robbers on Graphs
![Page 77: The Game of Cops and Robbers on Graphs - Inria · 2017-11-06 · 1/19 The Game of Cops and Robbers on Graphs Fionn Mc Inerney Universit e C^ote d’Azur, Inria, CNRS, I3S, France](https://reader034.vdocuments.us/reader034/viewer/2022042119/5e97c5c04161dd2ff77594ab/html5/thumbnails/77.jpg)
13/19
Link with Graph Structural Properties
Reminder: For any graph G , cn(G) ≤ γ(G) the dominating number of G .
Lemma [Aigner, Fromme 1984]
1 Cop is sufficient to “protect” a shortest path P in any graph.(after a finite number of steps, Robber cannot reach P)
⇒ cn(grid) = 2 (while γ(grid) ≈ n/5)
⇒ Cop-number related to structural properties
F. Mc Inerney The Game of Cops and Robbers on Graphs
![Page 78: The Game of Cops and Robbers on Graphs - Inria · 2017-11-06 · 1/19 The Game of Cops and Robbers on Graphs Fionn Mc Inerney Universit e C^ote d’Azur, Inria, CNRS, I3S, France](https://reader034.vdocuments.us/reader034/viewer/2022042119/5e97c5c04161dd2ff77594ab/html5/thumbnails/78.jpg)
13/19
Link with Graph Structural Properties
Reminder: For any graph G , cn(G) ≤ γ(G) the dominating number of G .
Lemma [Aigner, Fromme 1984]
1 Cop is sufficient to “protect” a shortest path P in any graph.(after a finite number of steps, Robber cannot reach P)
⇒ cn(grid) = 2 (while γ(grid) ≈ n/5)
⇒ Cop-number related to structural properties
F. Mc Inerney The Game of Cops and Robbers on Graphs
![Page 79: The Game of Cops and Robbers on Graphs - Inria · 2017-11-06 · 1/19 The Game of Cops and Robbers on Graphs Fionn Mc Inerney Universit e C^ote d’Azur, Inria, CNRS, I3S, France](https://reader034.vdocuments.us/reader034/viewer/2022042119/5e97c5c04161dd2ff77594ab/html5/thumbnails/79.jpg)
13/19
Link with Graph Structural Properties
Reminder: For any graph G , cn(G) ≤ γ(G) the dominating number of G .
Lemma [Aigner, Fromme 1984]
1 Cop is sufficient to “protect” a shortest path P in any graph.(after a finite number of steps, Robber cannot reach P)
⇒ cn(grid) = 2 (while γ(grid) ≈ n/5)
⇒ Cop-number related to structural properties
F. Mc Inerney The Game of Cops and Robbers on Graphs
![Page 80: The Game of Cops and Robbers on Graphs - Inria · 2017-11-06 · 1/19 The Game of Cops and Robbers on Graphs Fionn Mc Inerney Universit e C^ote d’Azur, Inria, CNRS, I3S, France](https://reader034.vdocuments.us/reader034/viewer/2022042119/5e97c5c04161dd2ff77594ab/html5/thumbnails/80.jpg)
14/19
1 Cop can protect 1 shortest path: applications (1)
For any planar graph G (there is a drawing of G on the plane without crossing edges),there exists separators consisting of ≤ 3 shortest paths
Cop-number vs. graph structure a surprising (?) example
cn(G) ≤ 3 for any planar graph G [Aigner and Fromme 84]
F. Mc Inerney The Game of Cops and Robbers on Graphs
![Page 81: The Game of Cops and Robbers on Graphs - Inria · 2017-11-06 · 1/19 The Game of Cops and Robbers on Graphs Fionn Mc Inerney Universit e C^ote d’Azur, Inria, CNRS, I3S, France](https://reader034.vdocuments.us/reader034/viewer/2022042119/5e97c5c04161dd2ff77594ab/html5/thumbnails/81.jpg)
14/19
1 Cop can protect 1 shortest path: applications (1)
For any planar graph G (there is a drawing of G on the plane without crossing edges),there exists separators consisting of ≤ 3 shortest paths
Cop-number vs. graph structure a surprising (?) example
cn(G) ≤ 3 for any planar graph G [Aigner and Fromme 84]
F. Mc Inerney The Game of Cops and Robbers on Graphs
![Page 82: The Game of Cops and Robbers on Graphs - Inria · 2017-11-06 · 1/19 The Game of Cops and Robbers on Graphs Fionn Mc Inerney Universit e C^ote d’Azur, Inria, CNRS, I3S, France](https://reader034.vdocuments.us/reader034/viewer/2022042119/5e97c5c04161dd2ff77594ab/html5/thumbnails/82.jpg)
14/19
1 Cop can protect 1 shortest path: applications (1)
For any planar graph G (there is a drawing of G on the plane without crossing edges),there exists separators consisting of ≤ 3 shortest paths
Cop-number vs. graph structure a surprising (?) example
cn(G) ≤ 3 for any planar graph G [Aigner and Fromme 84]
F. Mc Inerney The Game of Cops and Robbers on Graphs
![Page 83: The Game of Cops and Robbers on Graphs - Inria · 2017-11-06 · 1/19 The Game of Cops and Robbers on Graphs Fionn Mc Inerney Universit e C^ote d’Azur, Inria, CNRS, I3S, France](https://reader034.vdocuments.us/reader034/viewer/2022042119/5e97c5c04161dd2ff77594ab/html5/thumbnails/83.jpg)
15/19
1 Cop can protect 1 shortest path: applications (2)
G with genus ≤ g : can be drawn on a surface with ≤ g “handles”.
Cop-number vs. graph structure let’s go further
cn(G) ≤ b 3g2c+ 3 for any graph G with genus ≤ g [Schroder, 01]
Conjectures: cn(G) ≤ g + 3? cn(G) ≤ 3 if G has genus 1?
F. Mc Inerney The Game of Cops and Robbers on Graphs
![Page 84: The Game of Cops and Robbers on Graphs - Inria · 2017-11-06 · 1/19 The Game of Cops and Robbers on Graphs Fionn Mc Inerney Universit e C^ote d’Azur, Inria, CNRS, I3S, France](https://reader034.vdocuments.us/reader034/viewer/2022042119/5e97c5c04161dd2ff77594ab/html5/thumbnails/84.jpg)
15/19
1 Cop can protect 1 shortest path: applications (2)
G with genus ≤ g : can be drawn on a surface with ≤ g “handles”.
Cop-number vs. graph structure let’s go further
cn(G) ≤ b 3g2c+ 3 for any graph G with genus ≤ g [Schroder, 01]
Conjectures: cn(G) ≤ g + 3? cn(G) ≤ 3 if G has genus 1?
F. Mc Inerney The Game of Cops and Robbers on Graphs
![Page 85: The Game of Cops and Robbers on Graphs - Inria · 2017-11-06 · 1/19 The Game of Cops and Robbers on Graphs Fionn Mc Inerney Universit e C^ote d’Azur, Inria, CNRS, I3S, France](https://reader034.vdocuments.us/reader034/viewer/2022042119/5e97c5c04161dd2ff77594ab/html5/thumbnails/85.jpg)
16/19
1 Cop can protect 1 shortest path: applications (3)
s t
Lemma shortest-path-caterpillar = closed neighborhood of a shortest path [Chiniforooshan 2008]
5 Cops are sufficient to “protect” 1 shortest-path-caterpillar in any graph.
Any graph can be partitioned in n/ log n shortest-path-caterpillars (consider a BFS)
For any graph G , cn(G) = O(n/ log n) [Chiniforooshan 2008]
F. Mc Inerney The Game of Cops and Robbers on Graphs
![Page 86: The Game of Cops and Robbers on Graphs - Inria · 2017-11-06 · 1/19 The Game of Cops and Robbers on Graphs Fionn Mc Inerney Universit e C^ote d’Azur, Inria, CNRS, I3S, France](https://reader034.vdocuments.us/reader034/viewer/2022042119/5e97c5c04161dd2ff77594ab/html5/thumbnails/86.jpg)
16/19
1 Cop can protect 1 shortest path: applications (3)
s t
Lemma shortest-path-caterpillar = closed neighborhood of a shortest path [Chiniforooshan 2008]
5 Cops are sufficient to “protect” 1 shortest-path-caterpillar in any graph.Any graph can be partitioned in n/ log n shortest-path-caterpillars (consider a BFS)
For any graph G , cn(G) = O(n/ log n) [Chiniforooshan 2008]
F. Mc Inerney The Game of Cops and Robbers on Graphs
![Page 87: The Game of Cops and Robbers on Graphs - Inria · 2017-11-06 · 1/19 The Game of Cops and Robbers on Graphs Fionn Mc Inerney Universit e C^ote d’Azur, Inria, CNRS, I3S, France](https://reader034.vdocuments.us/reader034/viewer/2022042119/5e97c5c04161dd2ff77594ab/html5/thumbnails/87.jpg)
16/19
1 Cop can protect 1 shortest path: applications (3)
s t
Lemma shortest-path-caterpillar = closed neighborhood of a shortest path [Chiniforooshan 2008]
5 Cops are sufficient to “protect” 1 shortest-path-caterpillar in any graph.Any graph can be partitioned in n/ log n shortest-path-caterpillars (consider a BFS)
For any graph G , cn(G) = O(n/ log n) [Chiniforooshan 2008]
F. Mc Inerney The Game of Cops and Robbers on Graphs
![Page 88: The Game of Cops and Robbers on Graphs - Inria · 2017-11-06 · 1/19 The Game of Cops and Robbers on Graphs Fionn Mc Inerney Universit e C^ote d’Azur, Inria, CNRS, I3S, France](https://reader034.vdocuments.us/reader034/viewer/2022042119/5e97c5c04161dd2ff77594ab/html5/thumbnails/88.jpg)
17/19
Progress on Meyniel’s ConjectureMeyniel’s Conjecture [85]: For any n-node connected graph G , cn(G) = O(
√n)
cndominating set ≤ k ≤ k [folklore]
treewidth ≤ t ≤ t/2 + 1 [Joret, Kaminski,Theis 09]
chordality ≤ k < k [Kosowski,Li,Nisse,Suchan 12]
genus ≤ g ≤ b 3g2c+ 3 (conjecture ≤ g + 3) [Schroder, 01]
H-minor free ≤ |E(H)| [Andreae, 86]
degeneracy ≤ d ≤ d [Lu,Peng 12]
diameter 2 O(√n) −
bipartite diameter 3 O(√n) −
Erdos-Renyi graphs O(√n) [Bollobas et al. 08] [Luczak, Pralat 10]
Power law O(√n) [Bonato,Pralat,Wang 07]
A long story not finished yet...
cn(G) = O( nlog log n
) [Frankl 1987]
cn(G) = O( nlog n
) [Chiniforooshan 2008]
cn(G) = O( n
2(1−o(1))√
log n) [Scott, Sudakov 11, Lu,Peng 12]
note that n
2(1−o(1))√
log n≥ n1−ε for any ε > 0
F. Mc Inerney The Game of Cops and Robbers on Graphs
![Page 89: The Game of Cops and Robbers on Graphs - Inria · 2017-11-06 · 1/19 The Game of Cops and Robbers on Graphs Fionn Mc Inerney Universit e C^ote d’Azur, Inria, CNRS, I3S, France](https://reader034.vdocuments.us/reader034/viewer/2022042119/5e97c5c04161dd2ff77594ab/html5/thumbnails/89.jpg)
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Conclusion / Open problems
Meyniel’s Conjecture [1985]: For any n-node connected graph G , cn(G) = O(√n)
Conjecture [?]: For any n-node connected graph G with genus g , cn(G) ≤ g + 3
simpler(?) questions
cn(G) ≤ 3 if G has genus ≤ 1?
Directed graphs ?
Many other variants and questions... (e.g. [Clarke’09] [Bonato, et a.’13]...)
B. Alspach. Searching and sweeping graphs: a brief survey. In Le Matematiche, pages 5-37, 2004.W. Baird and A. Bonato. Meyniel’s conjecture on the cop number: a survey. http://arxiv.org/abs/1308.3385. 2013A. Bonato and R. J. Nowakowski. The game of Cops and Robber on Graphs. American Math. Soc., 2011.
F. Mc Inerney The Game of Cops and Robbers on Graphs
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19/19
Many Other Variants
Other VariantsVariant DifferencesFast Cops and Robbers Robber moves at speed sCops and Robbers at a Distance Robber can be captured at distance dLazy Cops and Robbers Only one cop may move each turnCops and Invisible Robbers Robber is invisibleCops and Robbers (capture time) Minimize capture time
Other Games on Graphs
Eternal Domination Game
Spy Game
Firefighting
Angel Problem
F. Mc Inerney The Game of Cops and Robbers on Graphs