1. On Tractable Parameterizations of Graph Isomorphism Adam Bouland, Anuj Dawar and Eryk Kopczyński

2. G H G 1 G 2 Is ?

3. What is the parameterized complexity of Graph Isomorphism?

4. Tree-Width Path-Width Tree-DepthMax Leaf Number Vertex Cover Number Size of smallest excluded minor Genus Crossing Number

5. Tree-Width Path-Width Tree-DepthMax Leaf Number Vertex Cover Number Size of smallest excluded minor Genus Crossing Number XP n f(k)

6. Tree-Width Path-Width Tree-DepthMax Leaf Number Vertex Cover Number Size of smallest excluded minor Genus Crossing Number FPT ? ? ? ? ? ? + Others f(k)n O(1)

7. Tree-Width Path-Width Tree-DepthMax Leaf Number Vertex Cover Number Size of smallest excluded minor Genus Crossing Number FPT ? ? ? ? ?

8. Tree-Width Path-Width Tree-DepthMax Leaf Number Vertex Cover Number Size of smallest excluded minor Genus Crossing Number FPT ? ? ? ? ? Generalized Tree-Depth

9. Why tree-depth? Theorem [Elberfeld Grohe Tantau 2012]: FO=MSO on a class of graphs C iff C has bounded tree-depth Game definition – similar to path-width Matrix factorization

10. Tree-Depth: 2 definitions “Closure” of ForestRooted Forest G has td(G)<=d iff G is a subgraph of the closure of a forest of depth d.

11. Proof Outline Decomposition Modify tree isomorphism algorithm Bound # vertices which can serve as root of decomposition

12. Proof Outline Decomposition Bound # vertices which can serve as root of decomposition Modify tree isomorphism algorithm

13. Tree-Depth: 2 definitions d cops1 robber Cop player wins if a cop lands on the robber

14. Tree-Depth: 2 definitions d cops1 robber

15. Tree-Depth: 2 definitions d cops1 robber

16. Tree-Depth: 2 definitions d cops1 robber

17. Tree-Depth: 2 definitions d cops1 robber

18. Tree-Depth: 2 definitions d cops1 robber

19. Tree-Depth: 2 definitions d cops1 robber

20. Tree-Depth: 2 definitions d cops1 robber Cop player wins if a cop lands on the robber

21. Tree-Depth: 2 definitions Fact: A graph has tree-depth d iff the Cop player has a winning strategy in the game using d cops

22. Tree-Depth: 2 definitions

25. Cop Wins

26. Bounding the Number of Roots Thm [Dvorak, Giannopolou and Thilikos 12]: The class C={G:td(G)≤d} is characterized by a finite set of forbidden subgraphs, each of size at most 2^2^(d-1) Cor: Number of roots of a graph of tree-depth d is at most 2^2^(d-1)

27. Bounding the Number of Roots H is forbidden subgraph for tree-depth <=d-1, and H has tree-depth d Cor: Number of roots of a graph of tree-depth d is at most 2^2^(d-1) G H

28. Bounding the Number of Roots SkSk S1S1 … S2S2 B

29. SkSk S1S1 … S2S2 B S i ≈S j iff there is an isomorphism from S i U B to S j U B which also preserves edges to B

30. Bounding the Number of Roots SkSk S1S1 … S2S2 B Thm: Deleting more than d copies of same component does not affect set of roots of the tree- depth

31. Bounding the Number of Roots SkSk S1S1 … S2S2 B Thm: Deleting more than d copies of same component does not affect set of roots of the tree- depth Idea: Never play cops in more than d copies Can “mirror” strategies using only d copies

32. Bounding the Number of Roots S1S1 S2S2 SkSk … G’ B S1S1 S1S1 SkSk S1S1 SkSk S1S1 S2S2 WLOG G is minimal #Vertices in component containing robber (and hence #Roots) bounded by reverse induction

33. Bounding the Number of Roots S1S1 S2S2 SkSk … G’ B S1S1 S1S1 SkSk S1S1 SkSk S1S1 S2S2 WLOG G is minimal #Vertices in component containing robber (and hence #Roots) bounded by reverse induction

34. Isomorphism Algorithm s Define S<T if 1.|S|<|T| 2.|S|=|T| and #s <#t 3.|S|=|T|, #s=#t. and (S 1 …S #s )<(T 1 …T #t ) where S_ i and T_ i are inductively ordered components of S and T

35. Isomorphism Algorithm Define S<T if 1.|S|<|T| 2.|S|=|T| and #s <#t 3.|S|=|T|, #s=#t and (E(s,r 1 )..E(s,r k ))< (E(t,r 1 )..E(t,r k )) 4. Above equal and (S 1 …S #s )<(T 1 …T #t ) s r1r1 Theorem 1: Graph Isomorphism is FPT in tree-depth

36. Extension: Subdivisions Defn: A graph has generalized tree-depth d iff it is a subdivision of a graph of tree- depth d Theorem 2: Graph Isomorphism is FPT in the generalized tree-depth

37. Tree-Width Path-Width Tree-DepthMax Leaf Number Vertex Cover Number Size of smallest excluded minor Genus Crossing Number FPT ? ? ? ? ? Generalized Tree-Depth

38. Questions ?