bandwidth, treewidth, separators, expansion, and universality filebandwidth, treewidth, separators,...
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Bandwidth, treewidth, separators, expansion, and universality
Bandwidth, treewidth, separators, expansion, anduniversality
Julia Bottcher
Technische Universitat Munchen
Topological & Geometric Graph Theory, 2008
(joint work with Klaas P. Pruessmann, Anusch Taraz & Andreas Wurfl)
Julia Bottcher TU Munchen TGGT ’08
Bandwidth, treewidth, separators, expansion, and universality
The bandwidth of planar graphs of bounded degreeBandwidth:
Let G be a graph on n vertices. Then bw(G) ≤ b if there is a labelling ofV (G) by 1, . . . , n s.t. for all {i, j} ∈ E(G) we have |i − j | ≤ b.
i j
Julia Bottcher TU Munchen TGGT ’08
Bandwidth, treewidth, separators, expansion, and universality
The bandwidth of planar graphs of bounded degreeBandwidth:
Let G be a graph on n vertices. Then bw(G) ≤ b if there is a labelling ofV (G) by 1, . . . , n s.t. for all {i, j} ∈ E(G) we have |i − j | ≤ b.
i j
Examples:
Hamiltonian cycle: bandwidth 2
Julia Bottcher TU Munchen TGGT ’08
Bandwidth, treewidth, separators, expansion, and universality
The bandwidth of planar graphs of bounded degreeBandwidth:
Let G be a graph on n vertices. Then bw(G) ≤ b if there is a labelling ofV (G) by 1, . . . , n s.t. for all {i, j} ∈ E(G) we have |i − j | ≤ b.
i j
Examples:
Hamiltonian cycle: bandwidth 2
Star: bandwidth ⌊(n − 1)/2⌋
Julia Bottcher TU Munchen TGGT ’08
Bandwidth, treewidth, separators, expansion, and universality
The bandwidth of planar graphs of bounded degreeBandwidth:
Let G be a graph on n vertices. Then bw(G) ≤ b if there is a labelling ofV (G) by 1, . . . , n s.t. for all {i, j} ∈ E(G) we have |i − j | ≤ b.
i j
Examples:
Hamiltonian cycle: bandwidth 2
Star: bandwidth ⌊(n − 1)/2⌋Grid: bandwidth
√n
Julia Bottcher TU Munchen TGGT ’08
Bandwidth, treewidth, separators, expansion, and universality
The bandwidth of planar graphs of bounded degreeBandwidth:
Let G be a graph on n vertices. Then bw(G) ≤ b if there is a labelling ofV (G) by 1, . . . , n s.t. for all {i, j} ∈ E(G) we have |i − j | ≤ b.
i j
Examples:
Hamiltonian cycle: bandwidth 2
Star: bandwidth ⌊(n − 1)/2⌋Grid: bandwidth
√n
Julia Bottcher TU Munchen TGGT ’08
Bandwidth, treewidth, separators, expansion, and universality
The bandwidth of planar graphs of bounded degreeTheorem CHUNG’88
Let T be a tree on n vertices and maximal degree ∆(T ) ≤ ∆. Then T hasbandwidth at most O(n/ log∆ n).
Examples:
Hamiltonian cycle: bandwidth 2
Star: bandwidth ⌊(n − 1)/2⌋Grid: bandwidth
√n
Julia Bottcher TU Munchen TGGT ’08
Bandwidth, treewidth, separators, expansion, and universality
The bandwidth of planar graphs of bounded degreeTheorem CHUNG’88
Let T be a tree on n vertices and maximal degree ∆(T ) ≤ ∆. Then T hasbandwidth at most O(n/ log∆ n).
Question: What about planar graphs?
Let ∆ be constant. Do planar graphs G on n vertices with ∆(G) ≤ ∆ havebandwidth o(n)?
Star: bandwidth ⌊(n − 1)/2⌋Grid: bandwidth
√n
Julia Bottcher TU Munchen TGGT ’08
Bandwidth, treewidth, separators, expansion, and universality
Bandwidth, treewidth, separators, and expansionTheorem B, PRUESSMANN, TARAZ, WUERFL
Let G be a hereditary class of graphs with max. degree ∆. Then t.f.a.e.:
All n-vertex G ∈ G have bandwidth o(n),
All n-vertex G ∈ G have treewidth o(n),
All n-vertex G ∈ G have(
o(n), 2/3)
-separators,
All n-vertex G ∈ G are(
o(n), ε)
-bounded.
Julia Bottcher TU Munchen TGGT ’08
Bandwidth, treewidth, separators, expansion, and universality
Bandwidth, treewidth, separators, and expansionTheorem B, PRUESSMANN, TARAZ, WUERFL
Let G be a hereditary class of graphs with max. degree ∆. Then t.f.a.e.:
All n-vertex G ∈ G have bandwidth o(n),
All n-vertex G ∈ G have treewidth o(n),
All n-vertex G ∈ G have(
o(n), 2/3)
-separators,
All n-vertex G ∈ G are(
o(n), ε)
-bounded.
Julia Bottcher TU Munchen TGGT ’08
Bandwidth, treewidth, separators, expansion, and universality
Bandwidth, treewidth, separators, and expansionTheorem B, PRUESSMANN, TARAZ, WUERFL
Let G be a hereditary class of graphs with max. degree ∆. Then t.f.a.e.:
All n-vertex G ∈ G have bandwidth o(n),
All n-vertex G ∈ G have treewidth o(n),
All n-vertex G ∈ G have(
o(n), 2/3)
-separators,
All n-vertex G ∈ G are(
o(n), ε)
-bounded.
Julia Bottcher TU Munchen TGGT ’08
Bandwidth, treewidth, separators, expansion, and universality
Bandwidth, treewidth, separators, and expansionTheorem B, PRUESSMANN, TARAZ, WUERFL
Let G be a hereditary class of graphs with max. degree ∆. Then t.f.a.e.:
All n-vertex G ∈ G have bandwidth o(n),
All n-vertex G ∈ G have treewidth o(n),
All n-vertex G ∈ G have(
o(n), 2/3)
-separators,
All n-vertex G ∈ G are(
o(n), ε)
-bounded.
tree decomposition
Julia Bottcher TU Munchen TGGT ’08
Bandwidth, treewidth, separators, expansion, and universality
Bandwidth, treewidth, separators, and expansionTheorem B, PRUESSMANN, TARAZ, WUERFL
Let G be a hereditary class of graphs with max. degree ∆. Then t.f.a.e.:
All n-vertex G ∈ G have bandwidth o(n),
All n-vertex G ∈ G have treewidth o(n),
All n-vertex G ∈ G have(
o(n), 2/3)
-separators,
All n-vertex G ∈ G are(
o(n), ε)
-bounded.
width
tree decomposition
Julia Bottcher TU Munchen TGGT ’08
Bandwidth, treewidth, separators, expansion, and universality
Bandwidth, treewidth, separators, and expansionTheorem B, PRUESSMANN, TARAZ, WUERFL
Let G be a hereditary class of graphs with max. degree ∆. Then t.f.a.e.:
All n-vertex G ∈ G have bandwidth o(n),
All n-vertex G ∈ G have treewidth o(n),
All n-vertex G ∈ G have(
o(n), 2/3)
-separators,
All n-vertex G ∈ G are(
o(n), ε)
-bounded.
Julia Bottcher TU Munchen TGGT ’08
Bandwidth, treewidth, separators, expansion, and universality
Bandwidth, treewidth, separators, and expansionTheorem B, PRUESSMANN, TARAZ, WUERFL
Let G be a hereditary class of graphs with max. degree ∆. Then t.f.a.e.:
All n-vertex G ∈ G have bandwidth o(n),
All n-vertex G ∈ G have treewidth o(n),
All n-vertex G ∈ G have(
o(n), 2/3)
-separators,
All n-vertex G ∈ G are(
o(n), ε)
-bounded.
Julia Bottcher TU Munchen TGGT ’08
Bandwidth, treewidth, separators, expansion, and universality
Bandwidth, treewidth, separators, and expansionTheorem B, PRUESSMANN, TARAZ, WUERFL
Let G be a hereditary class of graphs with max. degree ∆. Then t.f.a.e.:
All n-vertex G ∈ G have bandwidth o(n),
All n-vertex G ∈ G have treewidth o(n),
All n-vertex G ∈ G have(
o(n), 2/3)
-separators,
All n-vertex G ∈ G are(
o(n), ε)
-bounded.
≤ 23n
23n ≥ o(n)
Julia Bottcher TU Munchen TGGT ’08
Bandwidth, treewidth, separators, expansion, and universality
Bandwidth, treewidth, separators, and expansionTheorem B, PRUESSMANN, TARAZ, WUERFL
Let G be a hereditary class of graphs with max. degree ∆. Then t.f.a.e.:
All n-vertex G ∈ G have bandwidth o(n),
All n-vertex G ∈ G have treewidth o(n),
All n-vertex G ∈ G have(
o(n), 2/3)
-separators,
All n-vertex G ∈ G are(
o(n), ε)
-bounded.
G is (b, ε)-bounded:
For all G ′ ⊆ G on n ′ ≥ b verticesthere is U ⊆ V (G ′) with |U| ≤ n ′
2such that |N(U)| ≤ ε|U|.
G
Julia Bottcher TU Munchen TGGT ’08
Bandwidth, treewidth, separators, expansion, and universality
Bandwidth, treewidth, separators, and expansionTheorem B, PRUESSMANN, TARAZ, WUERFL
Let G be a hereditary class of graphs with max. degree ∆. Then t.f.a.e.:
All n-vertex G ∈ G have bandwidth o(n),
All n-vertex G ∈ G have treewidth o(n),
All n-vertex G ∈ G have(
o(n), 2/3)
-separators,
All n-vertex G ∈ G are(
o(n), ε)
-bounded.
G is (b, ε)-bounded:
For all G ′ ⊆ G on n ′ ≥ b verticesthere is U ⊆ V (G ′) with |U| ≤ n ′
2such that |N(U)| ≤ ε|U|.
G
G ′ ⊆ G
Julia Bottcher TU Munchen TGGT ’08
Bandwidth, treewidth, separators, expansion, and universality
Bandwidth, treewidth, separators, and expansionTheorem B, PRUESSMANN, TARAZ, WUERFL
Let G be a hereditary class of graphs with max. degree ∆. Then t.f.a.e.:
All n-vertex G ∈ G have bandwidth o(n),
All n-vertex G ∈ G have treewidth o(n),
All n-vertex G ∈ G have(
o(n), 2/3)
-separators,
All n-vertex G ∈ G are(
o(n), ε)
-bounded.
G is (b, ε)-bounded:
For all G ′ ⊆ G on n ′ ≥ b verticesthere is U ⊆ V (G ′) with |U| ≤ n ′
2such that |N(U)| ≤ ε|U|.
G
G ′ ⊆ G
U N(U)
Julia Bottcher TU Munchen TGGT ’08
Bandwidth, treewidth, separators, expansion, and universality
Bandwidth, treewidth, separators, and expansionTheorem B, PRUESSMANN, TARAZ, WUERFL
Let G be a hereditary class of graphs with max. degree ∆. Then t.f.a.e.:
All n-vertex G ∈ G have bandwidth o(n),
All n-vertex G ∈ G have treewidth o(n),
All n-vertex G ∈ G have(
o(n), 2/3)
-separators,
All n-vertex G ∈ G are(
o(n), ε)
-bounded.
Theorem LIPTON & TARJAN ’79
Planar graphs G on n vertices have (o(n), 2/3)-separators.
Julia Bottcher TU Munchen TGGT ’08
Bandwidth, treewidth, separators, expansion, and universality
Bandwidth, treewidth, separators, and expansionTheorem B, PRUESSMANN, TARAZ, WUERFL
Let G be a hereditary class of graphs with max. degree ∆. Then t.f.a.e.:
All n-vertex G ∈ G have bandwidth o(n),
All n-vertex G ∈ G have treewidth o(n),
All n-vertex G ∈ G have(
o(n), 2/3)
-separators,
All n-vertex G ∈ G are(
o(n), ε)
-bounded.
Theorem ALON, SEYMOUR, THOMAS ’90
F -minor free G on n vertices have (|F |3/2o(n), 2/3)-separators.
Julia Bottcher TU Munchen TGGT ’08
Bandwidth, treewidth, separators, expansion, and universality
Bandwidth versus path partition width
Path partition
A partition V (G) = V1 _∪ . . . _∪Vt of G such that edges of G only run withinthe Vi and between Vi and Vi+1.The width of this partition is maxi |Vi |, and ppw(G) is the minimal width ofa path partition of G.
Julia Bottcher TU Munchen TGGT ’08
Bandwidth, treewidth, separators, expansion, and universality
Bandwidth versus path partition width
Path partition
A partition V (G) = V1 _∪ . . . _∪Vt of G such that edges of G only run withinthe Vi and between Vi and Vi+1.The width of this partition is maxi |Vi |, and ppw(G) is the minimal width ofa path partition of G.
V1 V2 Vt
Julia Bottcher TU Munchen TGGT ’08
Bandwidth, treewidth, separators, expansion, and universality
Bandwidth versus path partition width
Path partition
A partition V (G) = V1 _∪ . . . _∪Vt of G such that edges of G only run withinthe Vi and between Vi and Vi+1.The width of this partition is maxi |Vi |, and ppw(G) is the minimal width ofa path partition of G.
V1 V2 Vt
Observation
For all G we have ppw(G) ≤ bw(G) ≤ 2 ppw(G).
Julia Bottcher TU Munchen TGGT ’08
Bandwidth, treewidth, separators, expansion, and universality
The proof
All n-vertex G = (V , E) ∈ G have(
o(n), 2/3)
-separators ⇒
All n-vertex G ∈ G have bandwidth o(n).
Julia Bottcher TU Munchen TGGT ’08
Bandwidth, treewidth, separators, expansion, and universality
The proof
All n-vertex G = (V , E) ∈ G have(
o(n), 2/3)
-separators ⇒
All n-vertex G ∈ G have bandwidth o(n).
Goal: Find path partition V = V1 _∪ . . . _∪Vt+1 of G with Vi = o(n).
Julia Bottcher TU Munchen TGGT ’08
Bandwidth, treewidth, separators, expansion, and universality
The proof
All n-vertex G = (V , E) ∈ G have(
o(n), 2/3)
-separators ⇒
All n-vertex G ∈ G have bandwidth o(n).
Goal: Find path partition V = V1 _∪ . . . _∪Vt+1 of G with Vi = o(n).
G
Julia Bottcher TU Munchen TGGT ’08
Bandwidth, treewidth, separators, expansion, and universality
The proof
All n-vertex G = (V , E) ∈ G have(
o(n), 2/3)
-separators ⇒
All n-vertex G ∈ G have bandwidth o(n).
Goal: Find path partition V = V1 _∪ . . . _∪Vt+1 of G with Vi = o(n).
G
Julia Bottcher TU Munchen TGGT ’08
Bandwidth, treewidth, separators, expansion, and universality
The proof
All n-vertex G = (V , E) ∈ G have(
o(n), 2/3)
-separators ⇒
All n-vertex G ∈ G have bandwidth o(n).
Goal: Find path partition V = V1 _∪ . . . _∪Vt+1 of G with Vi = o(n).
G
Julia Bottcher TU Munchen TGGT ’08
Bandwidth, treewidth, separators, expansion, and universality
The proof
All n-vertex G = (V , E) ∈ G have(
o(n), 2/3)
-separators ⇒
All n-vertex G ∈ G have bandwidth o(n).
Goal: Find path partition V = V1 _∪ . . . _∪Vt+1 of G with Vi = o(n).
G
S
R1 R2
Rt
Julia Bottcher TU Munchen TGGT ’08
Bandwidth, treewidth, separators, expansion, and universality
The proof
All n-vertex G = (V , E) ∈ G have(
o(n), 2/3)
-separators ⇒
All n-vertex G ∈ G have bandwidth o(n).
Goal: Find path partition V = V1 _∪ . . . _∪Vt+1 of G with Vi = o(n).
G
S
R1 R2
Rt
Bi := {v : v ∈ Rj , i < j,dist(v , S) = i}
Julia Bottcher TU Munchen TGGT ’08
Bandwidth, treewidth, separators, expansion, and universality
The proof
All n-vertex G = (V , E) ∈ G have(
o(n), 2/3)
-separators ⇒
All n-vertex G ∈ G have bandwidth o(n).
Goal: Find path partition V = V1 _∪ . . . _∪Vt+1 of G with Vi = o(n).
G
S
R ′
1 R ′
2
R ′
t
B Bi := {v : v ∈ Rj , i < j,dist(v , S) = i}
B :=⋃
Bi , R ′
j := Rj \ B
Julia Bottcher TU Munchen TGGT ’08
Bandwidth, treewidth, separators, expansion, and universality
The proof
All n-vertex G = (V , E) ∈ G have(
o(n), 2/3)
-separators ⇒
All n-vertex G ∈ G have bandwidth o(n).
Goal: Find path partition V = V1 _∪ . . . _∪Vt+1 of G with Vi = o(n).
G
S
R ′
1 R ′
2
R ′
t
B Bi := {v : v ∈ Rj , i < j,dist(v , S) = i}
B :=⋃
Bi , R ′
j := Rj \ B
|S|, |R ′
j | = o(n)
⇒ |Bi | ≤ ∆t |S| = o(n)
Julia Bottcher TU Munchen TGGT ’08
Bandwidth, treewidth, separators, expansion, and universality
The proof
All n-vertex G = (V , E) ∈ G have(
o(n), 2/3)
-separators ⇒
All n-vertex G ∈ G have bandwidth o(n).
Goal: Find path partition V = V1 _∪ . . . _∪Vt+1 of G with Vi = o(n).
G
S
R ′
1 R ′
2
R ′
t
B Bi := {v : v ∈ Rj , i < j,dist(v , S) = i}
B :=⋃
Bi , R ′
j := Rj \ B
|S|, |R ′
j | = o(n)
⇒ |Bi | ≤ ∆t |S| = o(n)
V1 := S
Julia Bottcher TU Munchen TGGT ’08
Bandwidth, treewidth, separators, expansion, and universality
The proof
All n-vertex G = (V , E) ∈ G have(
o(n), 2/3)
-separators ⇒
All n-vertex G ∈ G have bandwidth o(n).
Goal: Find path partition V = V1 _∪ . . . _∪Vt+1 of G with Vi = o(n).
G
S
R ′
1 R ′
2
R ′
t
B Bi := {v : v ∈ Rj , i < j,dist(v , S) = i}
B :=⋃
Bi , R ′
j := Rj \ B
|S|, |R ′
j | = o(n)
⇒ |Bi | ≤ ∆t |S| = o(n)
V1 := S
Vi+1 := R ′
i _∪Bi
Julia Bottcher TU Munchen TGGT ’08
Bandwidth, treewidth, separators, expansion, and universality
A consequence: universality for planar graphs
Theorem
Every n-vertex graph H with min. degree δ(H) ≥ (34 + γ)n contains every
n-vertex planar graph G with max. degree ∆.
Julia Bottcher TU Munchen TGGT ’08
Bandwidth, treewidth, separators, expansion, and universality
A consequence: universality for planar graphs
Theorem
∀γ > 0, ∆∃n0 ∀n ≥ n0
Every n-vertex graph H with min. degree δ(H) ≥ (34 + γ)n contains every
n-vertex planar graph G with max. degree ∆.
Julia Bottcher TU Munchen TGGT ’08
Bandwidth, treewidth, separators, expansion, and universality
A consequence: universality for planar graphs
Theorem
∀γ > 0, ∆∃n0 ∀n ≥ n0
Every n-vertex graph H with min. degree δ(H) ≥ (34 + γ)n contains every
n-vertex planar graph G with max. degree ∆.
Universality for Graphs of Bounded Bandwidth B,SCHACHT,TARAZ’08
For all k , ∆ ≥ 1, and γ > 0 exists n0 and β > 0 s.t. for all n ≥ n0
χ(G) = k , ∆(G) ≤ ∆, bw(G) ≤ βn
δ(H) ≥ ( k−1k + γ)n =⇒ H contains G
Julia Bottcher TU Munchen TGGT ’08
Bandwidth, treewidth, separators, expansion, and universality
A consequence: universality for planar graphs
Theorem
∀γ > 0, ∆∃n0 ∀n ≥ n0
Every n-vertex graph H with min. degree δ(H) ≥ (34 + γ)n contains every
n-vertex planar graph G with max. degree ∆.
Universality for Graphs of Bounded Bandwidth B,SCHACHT,TARAZ’08
For all k , ∆ ≥ 1, and γ > 0 exists n0 and β > 0 s.t. for all n ≥ n0
χ(G) = k , ∆(G) ≤ ∆, bw(G) ≤ βn
δ(H) ≥ ( k−1k + γ)n =⇒ H contains G
Theorem KUHN, OSTHUS, TARAZ 2005
Every n-vertex graph H with min. degree δ(H) ≥ (23 + γ)n contains some
n-vertex planar triangulation G.
Julia Bottcher TU Munchen TGGT ’08
Bandwidth, treewidth, separators, expansion, and universality
A consequence: universality for planar graphs
Theorem
∀γ > 0, ∆∃n0 ∀n ≥ n0
Every n-vertex graph H with min. degree δ(H) ≥ (23 + γ)n contains every
n-vertex planar graph G with max. degree ∆ and chromatic number 3.
Universality for Graphs of Bounded Bandwidth B,SCHACHT,TARAZ’08
For all k , ∆ ≥ 1, and γ > 0 exists n0 and β > 0 s.t. for all n ≥ n0
χ(G) = k , ∆(G) ≤ ∆, bw(G) ≤ βn
δ(H) ≥ ( k−1k + γ)n =⇒ H contains G
Theorem KUHN, OSTHUS, TARAZ 2005
Every n-vertex graph H with min. degree δ(H) ≥ (23 + γ)n contains some
n-vertex planar triangulation G.
Julia Bottcher TU Munchen TGGT ’08
Bandwidth, treewidth, separators, expansion, and universality
Concluding remarks
Let G be a hereditary class of graphs with max. degree ∆. Then
Bandwidth, treewidth, separators, expansion are sublinearlyequivalent for G.
Each of them implies that all H with δ(H) ≥ ( r−1r + γ)n are universal
for the class of r -chromatic graphs in G.
Julia Bottcher TU Munchen TGGT ’08
Bandwidth, treewidth, separators, expansion, and universality
Concluding remarks
Let G be a hereditary class of graphs with max. degree ∆. Then
Bandwidth, treewidth, separators, expansion are sublinearlyequivalent for G.
Each of them implies that all H with δ(H) ≥ ( r−1r + γ)n are universal
for the class of r -chromatic graphs in G.
Question
Let G be an r -chromatic expander on n vertices (i.e. ∀U ⊆ V (G) with|U| ≤ n
2 we have |N(U)| ≥ εU for some constant ε > 0).Is it true that there is an n-vertex H with δ(H) ≥ ( r−1
r + γ)n but G 6⊆ H forγ > 0 sufficiently small?
Julia Bottcher TU Munchen TGGT ’08
Bandwidth, treewidth, separators, expansion, and universality
Concluding remarks
Let G be a hereditary class of graphs with max. degree ∆. Then
Bandwidth, treewidth, separators, expansion are sublinearlyequivalent for G.
Each of them implies that all H with δ(H) ≥ ( r−1r + γ)n are universal
for the class of r -chromatic graphs in G.
Question
Let G be an r -chromatic expander on n vertices (i.e. ∀U ⊆ V (G) with|U| ≤ n
2 we have |N(U)| ≥ εU for some constant ε > 0).Is it true that there is an n-vertex H with δ(H) ≥ ( r−1
r + γ)n but G 6⊆ H forγ > 0 sufficiently small?
Merci beaucoup.Julia Bottcher TU Munchen TGGT ’08