On Tractable Parameterizations of Graph Isomorphism

Adam Bouland, Anuj Dawar and Eryk Kopczyński

G H

G1 G2Is ?

What is the parameterized complexity of Graph

Isomorphism?

Tree-Width

Path-Width

Tree-Depth Max Leaf Number

Vertex Cover Number

Size of smallest excluded minor

Genus

Crossing Number

Tree-Width

Path-Width

Tree-Depth Max Leaf Number

Vertex Cover Number

Size of smallest excluded minor

Genus

Crossing Number

XPnf(k)

Tree-Width

Path-Width

Tree-Depth Max Leaf Number

Vertex Cover Number

Size of smallest excluded minor

Genus

Crossing Number

FPT?

? ?

? ?

?

+ Others

f(k)nO(1)

Tree-Width

Path-Width

Tree-Depth Max Leaf Number

Vertex Cover Number

Size of smallest excluded minor

Genus

Crossing Number

FPT?

? ?

? ?

Tree-Width

Path-Width

Tree-Depth Max Leaf Number

Vertex Cover Number

Size of smallest excluded minor

Genus

Crossing Number

FPT?

? ?

? ?

Generalized Tree-Depth

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

“Closure” of ForestRooted 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

Tree-Depth: 2 definitions

d cops 1 robber

Tree-Depth: 2 definitions

d cops 1 robber

Tree-Depth: 2 definitions

d cops 1 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

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

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

Bounding the Number of Roots

SkS1 …S2

B

Bounding the Number of Roots

SkS1 …S2

BSi ≈Sj iff there is an isomorphism from Si U B to Sj U B which also preserves edges

to B

Bounding the Number of Roots

SkS1 …S2

BThm: Deleting more than d

copies of same component does not affect set of roots of the tree-

depth

Bounding the Number of Roots

SkS1 …S2

BThm: 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

Bounding the Number of Roots

S1 S2 Sk…

G’

B

S1S1 SkS1 SkS1 S2

WLOG G is minimal

#Vertices in component containing robber (and

hence #Roots) bounded by reverse induction

Bounding the Number of Roots

S1 S2 Sk…

G’

B

S1S1 SkS1 SkS1 S2

WLOG G is minimal

#Vertices in component containing robber (and

hence #Roots) bounded by reverse induction

Isomorphism Algorithm

s

Define S<T if

1. |S|<|T|

2. |S|=|T| and #s <#t

3. |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

Isomorphism AlgorithmDefine S<T if

1. |S|<|T|

2. |S|=|T| and #s <#t

3. |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)

s

r1

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 tree-depth d

Theorem 2: Graph Isomorphism is FPT in the generalized tree-depth

Tree-Width

Path-Width

Tree-Depth Max Leaf Number

Vertex Cover Number

Size of smallest excluded minor

Genus

Crossing Number

FPT?

? ?

? ?

Generalized Tree-Depth

Questions

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