On Tractable Parameterizations of Graph Isomorphism Adam Bouland

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 Path-Width Crossing Number Tree-Depth Max Leaf Number Vertex Cover Number

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

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

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

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

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

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.

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

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

Tree-Depth: 2 definitions d cops 1 robber Cop player wins if a cop lands on the robber

Tree-Depth: 2 definitions d cops 1 robber

Tree-Depth: 2 definitions d cops 1 robber Cop player wins if a cop lands on the robber

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

Tree-Depth: 2 definitions

Tree-Depth: 2 definitions Cop Wins

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)

Bounding the Number of Roots G H 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)

Bounding the Number of Roots B S 1 S 2 … Sk

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 S 1 S 2 … Sk

Bounding the Number of Roots Thm: Deleting more than d copies of same component does not affect set of roots of the treedepth B S 1 S 2 … Sk

Bounding the Number of Roots Thm: Deleting more than d copies of same component does not affect set of roots of the treedepth Idea: Never play cops in more than d copies B Can “mirror” strategies using only d copies S 1 S 2 … Sk

Bounding the Number of Roots B WLOG G is minimal G’ S 1 S 2 … Sk #Vertices in component containing robber (and hence #Roots) bounded by reverse induction

Isomorphism Algorithm Define S<T if s 1. |S|<|T| 2. |S|=|T| and #s <#t 3. |S|=|T|, #s=#t. and 4. (S 1…S#s)<(T 1…T#t) 5. where S_i and T_i are inductively ordered components of S and T

Isomorphism Algorithm Define S<T if 1. |S|<|T| r 1 2. |S|=|T| and #s <#t s 3. |S|=|T|, #s=#t and 4. (E(s, r 1). . E(s, rk))< (E(t, r 1). . E(t, rk)) 5. 4. Above equal and 6. (S 1…S#s)<(T 1…T#t) Theorem 1: Graph Isomorphism is FPT in tree-depth

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

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

Questions ?