1) The document discusses the game of cops and robbers played on oriented planar graphs. It provides background on the game and defines the problem of determining the cop number of a graph. 2) It then presents four results: (1) analyzing an existing AF strategy, (2) establishing a new lower bound using a modified AF strategy, (3) providing an upper bound using separator theorems, and (4) a new lower bound construction using shelter graphs. 3) The conclusions note that further reducing the upper bound and increasing the lower bound remains challenging and would require comprehensive cop cooperation strategies and generating robber-favorable graphs.

1. Cops and Robbers:
On Oriented Planar Graphs
Si Young Oh
Advisor: Professor. Po-Shen Loh
1

2. Overview
• Background:
• What is the game of cops and robbers?
• Issue:
• What is the problem? / What is known? / What do we want to
know?
• Four results
• Conclusion
2

3. Background
• Played by two distinct players.
• c cops and one robber
• Game is played on a graph G = (V, E).
• Perfect information game
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4. Background
1. Cops choose where to put c cops.
Multiple cops may be on the same vertex.
2. The robber chooses a vertex.
3. Subset of cops move along edges.
4. Robber can either stay or move along a edge.
5. Repeat 3 and 4.
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5. Background
• Cops win if one of cops is at the same vertex with the robber
at any moment.
• Robber wins if he can escape indefinitely.
• It’s like a simple version of Pac-Man!
5

6. Background
• Graphs can be either undirected or oriented.
• Directed graphs are in between.
• Only connected undirected graphs are interesting.
• Only strongly connected oriented graphs are interesting.
• Multiple edges or self-loops are ignored.
• Non-empty subset of cops are moving on cops’ turn.
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7. Issue
• What is the smallest c(G), cop number, such that c cops are
enough to win on G?
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8. Issue
• What is the smallest c(G), cop number, such that c cops are
enough to win on G?
2 2
8

9. Issue
• What is the smallest c(G), cop number, such that c cops are
enough to win on G?
2 2
When there’s only one cop, the robber can indefinitely escape. 9

10. Issue
• How about this?
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11. Issue
•
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12. Issue
• It is not trivial to find a cop number on a graph.
• It is not trivial to verify if c is a cop number on a graph.
• With c cops, is it possible for the cops to catch the robber for
every robber’s strategy?
• With (c - 1) cops, is it possible for the robber to escape
indefinitely for every cops’ strategy.
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13. Issue
•
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14. Issue
•
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15. Issue
•
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16. Issue
• Four goals of the talk:
1. Study known strategies to see advantages and disadvantages
focusing on the AF strategy.
2. Establish lower bound on general oriented graphs by modifying
the AF strategy.
3. Establish upper bounds on planar oriented graphs using Lipton
and Tarjan’s separator theorem.
4. Establish lower bounds on planar oriented graphs that does
better than the modified AF strategy. 16

17. Result (1) – AF strategy
• Why does it work?
• What happens when there are k – 1 cops?
1
2
d
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18. Result (1) – AF strategy
• The robber simply stays when none of cops can catch the
robber immediately.
• The neighbor v is not available when 1
• A cop is at v 2
• A cop is at a neighbor of v
• One cop can block at most one
neighbor.
d
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19. Result (1) – AF strategy
•
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20. Result (1) – AF strategy
•
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21. Result (1) – AF strategy
• Is there some nice properties about 3- or 4-cycles?
• We first suspected that 3-cycles are good for cops.
• Both cops and a robber has same advantage when moving.
• Cops can guard vertices more efficiently.
• For example, a 4-cycle has cop number 2 but adding a diagonal
edge reduce the cop number to 1.
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22. Result (1) – AF strategy
• How does 3-cycles affect the cop number?
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23. Result (1) – AF strategy
•
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24. Result (1) – AF strategy
• Cop number is at most 3 on planar undirected graphs proven
by Aigner and Fromme.
• The proof uses an algorithm to block a shortest path.
• Two disjoint paths separate a graph into two parts.
• Three cops can cooperate efficiently.
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25. Result (1) – AF strategy
• Using the AF strategy, we can find a planar undirected graph
such that the cop number is at least 3.
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26. Result (1) – AF strategy
• Conclusions about AF strategy on undirected graphs:
• Strong strategy to prove that cop number is higher than certain
numbers.
• It finds the best bound (known so far) on both general and planar
undirected graphs.
• Interesting to note that AF strategy on planar graphs cannot show
that cop number is higher than 3 because of Euler formula.
• Does it always find the best bound?
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27. Result (2) – Modified AF
•
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28. Result (2) – Modified AF
• Why does it work?
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29. Result (2) – Modified AF
•
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30. Result (2) – Modified AF
• On oriented graphs, the shortest path argument does not hold
anymore.
• So, the theorem that the cop number is at most 3 on planar
undirected graphs does not hold on planar oriented graphs.
• We need to establish new bounds.
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31. Result (3) – Upper bound
•
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32. Result (3) – Upper bound
•
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33. Result (4) – Lower bound
• Because of Euler formula, the modified AF strategy cannot
show that cop number is higher than 3.
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34. Result (4) – Lower bound
• Icosahedron
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35. Result (4) – Lower bound
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36. Result (4) – Lower bound
• Icosahedron → Truncated Icosahedron
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37. Result (4) – Lower bound
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38. Result (4) – Lower bound
• Icosahedron → Truncated Icosahedron → Great
Rhombicosidodecahedron
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39. Result (4) – Lower bound
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40. Result (4) – Lower bound
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41. Result (4) – Lower bound
• Icosahedron → Truncated Icosahedron → Great
Rhombicosidodecahedron → Put shelter
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42. Result (4) – Lower bound
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43. Result (4) – Lower bound
• Icosahedron → Truncated Icosahedron → Great
Rhombicosidodecahedron → Put shelter → Give directions
(part 1)
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44. Result (4) – Lower bound
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45. Result (4) – Lower bound
• Icosahedron → Truncated Icosahedron → Great
Rhombicosidodecahedron → Put shelter → Give directions
(part 1) → Give directions (part 2)
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46. Result (4) – Lower bound
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47. Result (4) – Lower bound
• Icosahedron → Truncated Icosahedron → Great
Rhombicosidodecahedron → Put shelter → Give directions
(part 1) → Give directions (part 2) → Final Graph.
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48. Result (4) – Lower bound
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49. Result (4) – Lower bound
• What happens when there are three cops?
• Initial:
• There are 20 shelters and one cop can block at most one shelter.
• The robber can choose a safe shelter.
• The robber waits until one cop is threatening.
• Three possible scenarios.
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50. Result (4) – Lower bound
• Scenario 1. All three cops are on the same unit.
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51. Result (4) – Lower bound
• Scenario 2. Two cops are on the same unit.
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52. Result (4) – Lower bound
• Scenario 3. Only one cop is on the same unit.
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53. Result (4) – Lower bound
•
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54. Conclusions
• It is challenging to reduce the upper bound.
• Since independent cops are not strong enough, we should find
comprehensive rules of cooperation on all the graphs.
• It is challenging to increase the lower bound.
• We should find a rule to generate robber-favorable graphs as the
number of vertices increase indefinitely.
• We should show all the cops’ strategies do not work.
• Still, the result shows that it might be possible to make a
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better robber’s strategy on non-planar graphs.

55. Conclusions
• Questions?
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