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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
Fact: A graph has tree-depth d iff the Cop player has a winning strategy in the game using d cops
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
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