On Tractable Parameterizations
of Graph Isomorphism
Adam Bouland, Anuj Dawar
and Eryk Kopczyński
G
H
Is G1  G2?
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
Size of smallest
excluded minor
XP
nf(k)
Tree-Width
Genus
Path-Width
Crossing Number
Tree-Depth
Max Leaf Number
Vertex Cover Number
FPT
Size of smallest
excluded minor
Tree-Width
?
Path-Width
?
O(1)
f(k)n
?
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
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
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
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
S1
S2
…
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
S1
S2
…
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
S1
S2
…
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
S1
S2
…
Sk
Bounding the Number of Roots
B
WLOG G is minimal
G’
S1
S2
…
Sk
#Vertices in component
containing robber (and
hence #Roots) bounded by
reverse induction
Bounding the Number of Roots
B
WLOG G is minimal
G’
S1
S2
…
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
(S1…S#s)<(T1…T#t)
where S_i and T_i are
inductively ordered
components of S and T
Isomorphism Algorithm
Define S<T if
1. |S|<|T|
r1
2. |S|=|T| and #s <#t
s
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)
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
Tree-Depth
Max Leaf Number
Vertex Cover Number
?
Questions
?