1. Cops and Robber on Planar Graphs Presented by Aaron Maurer Results of Work Done at IMA REU 2010 Aaron Maurer (Carleton College), John McCauley (Haverford College), and Silviya Valeva (Mount Holyoke College) Advisor: Andrew Beveridge (Macalester College) Problem Poser: Volkan Isler (University of Minnesota) Mentor: Vishal Saraswat (University of Minnesota)

2. What is Cops and Robbers Game? ● It is played on a graph ● One robber and n cops are placed on vertices. ● The robber and all the cops together take turns moving ● For a move, a cop/robber can either stay on the vertex he is on, or move to one which shares an edge with it ● If a cop ends up on the same vertex as the robber, they capture him and the cops win

3. Terminology ● N Cop Win: A graph G is called n cop win if n cops, for any initial arraignment of cops and robber, can always capture the robber ● C(G) (“cop number”) is smallest n such that this is true ● N(v) (“neighborhood of v”) is the set of vertices that are or share an edge with a vertex v ● Dominating Set: a set of vertices such that all vertices in a neighborhood or graph shares a edge with one of the vertices

4. Planar Graphs ● Planar: A planar graph is a graph who can be drawn in a plane such that its edges only meet at vertices (no edge crossings) ● “A Game of Cops and Robbers” by Aigner and Fromme provided several important theorems for games of cops and robbers on planar graphs.

5. Aigner and Fromme ● Pitfall: A “pitfall” is a vertex whose entire neighborhood is dominated by a single vertex ● Theorem: A graph G is one cop win if and only if by successively removing pit falls in any order, G is reduced to a single vertex. ● Proof Outline: If a robber is on a pitfall, cop can pretend he is on dominating vertex instead (from their robber has no where to go where the cop can't catch him immediately)

6. Aigner and Fromme ● Theorem: If given enough turns, a cop can position himself on a shortest path between two points such that if the robber moves onto it, the cop will capture him. ● Proof Outline: Since the path is shortest, any path between points on it is shortest, so cop can move to position where he can reach any other point on path quicker than robber.

7. Aigner and Fromme ● Big Theorem: For all planar graphs G, C(G) is at most 3. ● Proof Outline: ● -Two cops can choose shortest paths that share endpoints, dividing graph in two. ● -Third cop can choose another shortest path that reduces robber's section and frees another cop. ● -If repeated long enough, robber runs out of space and is caught.

8. Where Does This Leave Us ● A one cop win graph is easily identifiable ● At worst, a planar graph is three cop win ● Our Question: Can we categorize which planar graphs are two cop win and which are three cop win?

9. Examples: Platonic Solids Tetrahedron ● One cop win: dominated by a single vertex

10. Examples: Platonic Solids Cube ● Two vertices dominate ● No Pitfalls ● Thus, two cop win

11. Examples: Platonic Solids Dodecahedron ● No Pitfalls ● Whenever two cops approach robber, he always has escape ● Three cop win

12. Maximal Planar Graphs ● Maximal Planar: A graph is maximal planar when any additional edge would force the graph to be non-planar. ● This is also known as triangulated since it forces every face to have three edges.

13. Maximal Planar With Maximum Degree Less Than Five ● Theorem: If a maximal planar graph only has vertices of degree five or less, then it is at most two cop ● Proof Outline: ● -The cops can choose paths to the robber through two different vertices incident to him ● -As robber moves, cops can adjust their paths and move down them such that their combined is regularly decreasing ● -Once combined lengths paths get small enough, robber caught

14. Maximal Planar with Maximum Degree Less Than Five

15. Maximal Planar With Maximum Degree Greater Than Seven ● We have found examples of maximum planar graphs maximum degree seven that are cop number three

16. Example: Maximal Planar Maximum Degree 9 ● Created by adding extra vertices and edges in each face of dodecahedron as such:

17. Maximal Planar Maximum Degree Six ● Unknown if these can be three cop win. Our conjecture is that it can't ● Reasons: ● -The cops can always at least maintain their distance and paths while robber goes over degree six vertices ● -Always at worst two cop win unless there is a cycle of degree six vertices

18. Series Parallel Graphs ● If two graphs G and H have a source and a sink: ● Series Composition: combining one source with other sink is a ● Parallel Composition: combining both sources and both sinks is a ● Series-Parallel Graph: A graph made by a sequence of such compositions on a set of single edge graphs

19. Series Parallel Graphs

20. ● Theorem: All series-parallel graphs have cop number at most two. ● Proof Outline: ● -One cop goes to source, other to sink. ● -If parallel composition, robber stuck in one subgraph ● -If series, both cops can move towards merged source/sink till one gets there without letting robber escape from current subgraph. ● -repeat on successive SP subgraphs till caught

21. K-Outerplanar Graphs ● Outerplanar: a graph such that vertices lie on a fixed circle and all edges lie inside (also known as 1-outerplanar) ● K-outerplanar: a graph such that if the vertices on the exterior face is removed, one gets a (k-1)- outerplanar graph ● Maximal k-outerplanar: a k-outerplanar graph where the addition of another edge would make it non k-outerplanar

22. K-Outerplanar Graphs

23. Outerplanar Graphs ● Theorem: All outerplanar graphs are at most two cop win. ● Outline Proof: ● -Let cops move to either end of chord of graph, dividing graph ● -One cop moves to edge of next chord on robber side ● -Robber either on path between two chords or stuck in smaller area ● -Can be repeated till robber captured ● Better proof: Outerplanar is also series parallel

24. Maximal Outerplanar ● Theorem: All maximal outerplanar graphs are one cop win ● Outline Proof: Each maximal outerplanar graph has at least two pitfalls. Once removed, graph still maximal outerplanar. Can be repeated till one point left

25. Maximal 2-Outerplanar ● Theorem: Maximal 2-Outerplanar graphs have at most cop number 2 ● Outline Proof: Due to being maximal, every vertex inner circle connected to one on outer. ● -Cops can choose chord through middle and defend paths upwards from it, pinning robber in smaller space ● -Can systematically reduce size this area

26. Outerplanar ● Theorem: Maximal 3-outerplanar at most cop number 2 (proof similar 2-outerplanar but with more cases) ● Maximal k-outerplanar for k>3 unknown, we suspect at worst two cop win ● K-outerplanar in general for k>1 unknown

27. Conclusion ● Maximal Planar with maximum degree five at most two cop, maximum degree seven or more can be three cop, maximum degree six unknown ● Series-parallel graphs are at most two cop ● Outerplanar at most two cop, k-outerplanar unknown ● ● Maximal k-outerplanar for k 3 unknown

28. For Further Information If you have any further questions or want to see my REU groups report, with far more detail and more many more results, email me at: maurera@carleton.edu