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On Tractable Parameterizations of Graph Isomorphism

On Tractable Parameterizations of Graph Isomorphism. Adam Bouland, Anuj Dawar and Eryk Kopczyński. G. H. Is ?. G 1 G 2. What is the parameterized complexity of Graph Isomorphism?. Size of smallest excluded minor. Tree-Width. Genus. Crossing Number. Path-Width.

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On Tractable Parameterizations of Graph Isomorphism

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

  2. G H Is ? G1 G2

  3. What is the parameterized complexity of Graph Isomorphism?

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

  5. Size of smallest excluded minor XP Tree-Width Genus nf(k) Crossing Number Path-Width Tree-Depth Max Leaf Number Vertex Cover Number

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

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

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

  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 Rooted Forest “Closure” of 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 cops 1 robber Cop player wins if a cop lands on the robber

  14. Tree-Depth: 2 definitions d cops 1 robber

  15. Tree-Depth: 2 definitions d cops 1 robber

  16. Tree-Depth: 2 definitions d cops 1 robber

  17. Tree-Depth: 2 definitions d cops 1 robber

  18. Tree-Depth: 2 definitions d cops 1 robber

  19. Tree-Depth: 2 definitions d cops 1 robber

  20. Tree-Depth: 2 definitions d cops 1 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

  23. Tree-Depth: 2 definitions

  24. Tree-Depth: 2 definitions

  25. Tree-Depth: 2 definitions 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 G 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)

  28. Bounding the Number of Roots B S2 S1 … Sk

  29. Bounding the Number of Roots Si ≈Sj iff there is an isomorphism from Si U B to Sj U B which also preserves edges to B B S2 S1 … Sk

  30. Bounding the Number of Roots Thm: Deleting more than d copies of same component does not affect set of roots of the tree-depth B S2 S1 … Sk

  31. Bounding the Number of Roots 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 B S2 S1 … Sk

  32. Bounding the Number of Roots B WLOG G is minimal #Vertices in component containing robber (and hence #Roots) bounded by reverse induction G’ S1 S1 S1 S1 S1 S2 S2 … Sk Sk Sk

  33. Bounding the Number of Roots B WLOG G is minimal #Vertices in component containing robber (and hence #Roots) bounded by reverse induction G’ S1 S1 S1 S1 S1 S2 S2 … Sk Sk Sk

  34. Isomorphism Algorithm • Define S<T if • |S|<|T| • |S|=|T| and #s <#t • |S|=|T|, #s=#t. and • (S1…S#s)<(T1…T#t) • where S_i and T_i are inductively ordered components of S and T s

  35. Isomorphism Algorithm • Define S<T if • |S|<|T| • |S|=|T| and #s <#t • |S|=|T|, #s=#t and • (E(s,r1)..E(s,rk))< (E(t,r1)..E(t,rk)) • 4.Above equal and • (S1…S#s)<(T1…T#t) r1 s 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. Size of smallest excluded minor ? FPT ? ? Tree-Width Genus ? ? Crossing Number Path-Width Generalized Tree-Depth Tree-Depth Max Leaf Number Vertex Cover Number

  38. Questions ?

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