Three Edge Lengths Suffice For Drawing Outerplanar Graphs Noga Alon - - PowerPoint PPT Presentation

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Three Edge Lengths Suffice For Drawing Outerplanar Graphs Noga Alon - - PowerPoint PPT Presentation

Oct 07, 2023 150 likes •718 views

Preliminaries Results Approaching the Problem Remarks and Open Problems Three Edge Lengths Suffice For Drawing Outerplanar Graphs Noga Alon Ohad N. Feldheim Tel-Aviv University IMU, 2012 Noga Alon and Ohad N. Feldheim Three Edge Lengths

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SLIDE 1

Preliminaries Results Approaching the Problem Remarks and Open Problems

Three Edge Lengths Suffice For Drawing Outerplanar Graphs

Noga Alon Ohad N. Feldheim

Tel-Aviv University

IMU, 2012

Noga Alon and Ohad N. Feldheim Three Edge Lengths Suffice For Drawing Outerplanar Graphs

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Preliminaries Results Approaching the Problem Remarks and Open Problems Embedding in the Plane Drawings Degenerate Drawings

Mapping Graphs to the Plane

Let G = (VG, EG), π : VG → C, v4 v3 v2 v1 v0 Graph G v4, v3 v2 v1 v0 π(G)

→

Noga Alon and Ohad N. Feldheim Three Edge Lengths Suffice For Drawing Outerplanar Graphs

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SLIDE 3

Preliminaries Results Approaching the Problem Remarks and Open Problems Embedding in the Plane Drawings Degenerate Drawings

Mapping Graphs to the Plane

Let G = (VG, EG), π : VG → C, e = (v0, v1). We set: π(e) := (π(v0), π(v1)) v4 v3 v2 v1 v0 Graph G v4, v3 v2 v1 v0 π(G)

→

Noga Alon and Ohad N. Feldheim Three Edge Lengths Suffice For Drawing Outerplanar Graphs

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SLIDE 4

Preliminaries Results Approaching the Problem Remarks and Open Problems Embedding in the Plane Drawings Degenerate Drawings

Mapping Graphs to the Plane

Let G = (VG, EG), π : VG → C, e = (v0, v1). We set: π(e) := (π(v0), π(v1)) lenπ(e) := |π(v0) − π(v1)| v4 v3 v2 v1 v0 Graph G v4, v3 v2 v1 v0 π(G)

→

Noga Alon and Ohad N. Feldheim Three Edge Lengths Suffice For Drawing Outerplanar Graphs

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SLIDE 5

Preliminaries Results Approaching the Problem Remarks and Open Problems Embedding in the Plane Drawings Degenerate Drawings

Mapping Graphs to the Plane

Let G = (VG, EG), π : VG → C, e = (v0, v1). We set: π(e) := (π(v0), π(v1)) lenπ(e) := |π(v0) − π(v1)| lens(π) := |{lenπ(e) : e ∈ EG}| v4 v3 v2 v1 v0 Graph G v4, v3 v2 v1 v0 π(G)

→

Noga Alon and Ohad N. Feldheim Three Edge Lengths Suffice For Drawing Outerplanar Graphs

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SLIDE 6

Preliminaries Results Approaching the Problem Remarks and Open Problems Embedding in the Plane Drawings Degenerate Drawings

Drawings

Let G = (VG, EG), π : VG → C, Drawing: ∀v, v ′ ∈ VG, ∀e ∈ EG: π(v) = π(v ′), π(v) / ∈ π(e) v4 v3 v2 v1 v0 Graph G v4 v3 v2 v1 v0 π(G)

→

Noga Alon and Ohad N. Feldheim Three Edge Lengths Suffice For Drawing Outerplanar Graphs

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SLIDE 7

Preliminaries Results Approaching the Problem Remarks and Open Problems Embedding in the Plane Drawings Degenerate Drawings

Drawings

Let G = (VG, EG), π : VG → C, Drawing: ∀v, v ′ ∈ VG, ∀e ∈ EG: π(v) = π(v ′), π(v) / ∈ π(e) Distance Number dn(G) := min{lens(π) : π is a drawing of G} v4 v3 v2 v1 v0 Graph G v4 v3 v2 v1 v0 π(G)

→

Noga Alon and Ohad N. Feldheim Three Edge Lengths Suffice For Drawing Outerplanar Graphs

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SLIDE 8

Preliminaries Results Approaching the Problem Remarks and Open Problems Embedding in the Plane Drawings Degenerate Drawings

Degenerate Drawings

Let G = (VG, EG), π : VG → C,

  • Deg. Drawing: ∀v, v ′ ∈ VG: π(v) = π(v ′)

v4 v3 v2 v1 v0 Graph G v4 v3 v2 v1 v0 π(G)

→

Noga Alon and Ohad N. Feldheim Three Edge Lengths Suffice For Drawing Outerplanar Graphs

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SLIDE 9

Preliminaries Results Approaching the Problem Remarks and Open Problems Embedding in the Plane Drawings Degenerate Drawings

Degenerate Drawings

Let G = (VG, EG), π : VG → C,

  • Deg. Drawing: ∀v, v ′ ∈ VG: π(v) = π(v ′)

Degenerate Distance Number ddn(G) := min{lens(π) : π is a deg. drawing of G} v4 v3 v2 v1 v0 Graph G v4 v3 v2 v1 v0 π(G)

→

Noga Alon and Ohad N. Feldheim Three Edge Lengths Suffice For Drawing Outerplanar Graphs

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SLIDE 10

Preliminaries Results Approaching the Problem Remarks and Open Problems Embedding in the Plane Drawings Degenerate Drawings

Properties of the Distance Number

Properties of dn and ddn Let G ⊂ H. dn(G) ≤ dn(H) ddn(G) ≤ ddn(H) ddn(G) ≤ dn(G)

Noga Alon and Ohad N. Feldheim Three Edge Lengths Suffice For Drawing Outerplanar Graphs

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SLIDE 11

Preliminaries Results Approaching the Problem Remarks and Open Problems Previous Results Our Results

Distance Number - Previous Results

What bounds can we get on dn, ddn? Are they ever different?

Noga Alon and Ohad N. Feldheim Three Edge Lengths Suffice For Drawing Outerplanar Graphs

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SLIDE 12

Preliminaries Results Approaching the Problem Remarks and Open Problems Previous Results Our Results

Distance Number - Previous Results

What bounds can we get on dn, ddn? Are they ever different? Results on Kn Guth-Katz(’11)/Erd˝

  • s(’46):

c1n log n ≤ ddn(Kn) ≤ c2n √log n

Noga Alon and Ohad N. Feldheim Three Edge Lengths Suffice For Drawing Outerplanar Graphs

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SLIDE 13

Preliminaries Results Approaching the Problem Remarks and Open Problems Previous Results Our Results

Distance Number - Previous Results

What bounds can we get on dn, ddn? Are they ever different? Results on Kn Guth-Katz(’11)/Erd˝

  • s(’46):

c1n log n ≤ ddn(Kn) ≤ c2n √log n

Szemer´ edi(’95)/Erd˝

  • s(’51):

n−1 3

≤ dn(Kn) ≤ n

2

Noga Alon and Ohad N. Feldheim Three Edge Lengths Suffice For Drawing Outerplanar Graphs

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SLIDE 14

Preliminaries Results Approaching the Problem Remarks and Open Problems Previous Results Our Results

Distance Number - Previous Results

What bounds can we get on dn, ddn? Are they ever different? Results on Kn Guth-Katz(’11)/Erd˝

  • s(’46):

c1n log n ≤ ddn(Kn) ≤ c2n √log n

Szemer´ edi(’95)/Erd˝

  • s(’51):

n−1 3

≤ dn(Kn) ≤ n

2

Results on graphs with bdd. degree ∆, n vertices Carmi, Dujmovi´ c, Morin, Wood(’08): For ∆ ≥ 5: ddn is not uniformly bdd. For ∆ ≥ 7: exist Gn with ddn(G) = Ω(nc) for c(∆) < C < 1. dn(G) = O(∆4 log n) if Gn’s treewidth is uniformly bdd.

Noga Alon and Ohad N. Feldheim Three Edge Lengths Suffice For Drawing Outerplanar Graphs

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Preliminaries Results Approaching the Problem Remarks and Open Problems Previous Results Our Results

Distance Number - Planar Graphs

Planar Graph: Has a drawing without crossings.

Noga Alon and Ohad N. Feldheim Three Edge Lengths Suffice For Drawing Outerplanar Graphs

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Preliminaries Results Approaching the Problem Remarks and Open Problems Previous Results Our Results

Distance Number - Planar Graphs

Planar Graph: Has a drawing without crossings. ∃ planar Gn with ddn(G) = Ω(√n).

Noga Alon and Ohad N. Feldheim Three Edge Lengths Suffice For Drawing Outerplanar Graphs

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SLIDE 17

Preliminaries Results Approaching the Problem Remarks and Open Problems Previous Results Our Results

Distance Number - Planar Graphs

Planar Graph: Has a drawing without crossings. ∃ planar Gn with ddn(G) = Ω(√n). Carmi, Dujmovic, Morin, Wood(’08): Do outerplanar graphs have uniformly bounded (degenerate) distance number?

Noga Alon and Ohad N. Feldheim Three Edge Lengths Suffice For Drawing Outerplanar Graphs

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Preliminaries Results Approaching the Problem Remarks and Open Problems Previous Results Our Results

Outerplanar Graphs

Outerplanar Graph: ∃ drawing, without crossings, s.t. unbounded face contains all vertices. Outerplanar graph

Noga Alon and Ohad N. Feldheim Three Edge Lengths Suffice For Drawing Outerplanar Graphs

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SLIDE 19

Preliminaries Results Approaching the Problem Remarks and Open Problems Previous Results Our Results

Outerplanar Graphs

Outerplanar Graph: ∃ drawing, without crossings, s.t. unbounded face contains all vertices. Triangulated Outerplanar Graph: Outerplanar graph where all bounded faces are triangles. Outerplanar graph Trinagulated outerplanar graph

Noga Alon and Ohad N. Feldheim Three Edge Lengths Suffice For Drawing Outerplanar Graphs

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Preliminaries Results Approaching the Problem Remarks and Open Problems Previous Results Our Results

Our Results

Carmi, Dujmovic, Morin, Wood(’08): Do outerplanar graphs have uniformly bounded (degenerate) distance number?

Noga Alon and Ohad N. Feldheim Three Edge Lengths Suffice For Drawing Outerplanar Graphs

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Preliminaries Results Approaching the Problem Remarks and Open Problems Previous Results Our Results

Our Results

Carmi, Dujmovic, Morin, Wood(’08): Do outerplanar graphs have uniformly bounded (degenerate) distance number? Theorem (Alon, F.) For almost every triple a, b, c ∈ (0, 1), every outerplanar graph has a degenerate drawing using only edge-lengths a, b and c.

Noga Alon and Ohad N. Feldheim Three Edge Lengths Suffice For Drawing Outerplanar Graphs

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Preliminaries Results Approaching the Problem Remarks and Open Problems Previous Results Our Results

Our Results

Carmi, Dujmovic, Morin, Wood(’08): Do outerplanar graphs have uniformly bounded (degenerate) distance number? Theorem (Alon, F.) For almost every triple a, b, c ∈ (0, 1), every outerplanar graph has a degenerate drawing using only edge-lengths a, b and c. Work in progress... For almost every nine values a0, ..., a8 ∈ (0, 1), every outerplanar graph has a drawing using only edge-lengths a0, ..., a8.

Noga Alon and Ohad N. Feldheim Three Edge Lengths Suffice For Drawing Outerplanar Graphs

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SLIDE 23

Drawing Outerplanar Graphs

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Preliminaries Results Approaching the Problem Remarks and Open Problems Triangle Trees Naive Approach Embedding Rhombi The Construction

Triangle Trees

  • Bdd. faces of triangulated outerplanar graphs have a binary tree

structure.

Noga Alon and Ohad N. Feldheim Three Edge Lengths Suffice For Drawing Outerplanar Graphs

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Preliminaries Results Approaching the Problem Remarks and Open Problems Triangle Trees Naive Approach Embedding Rhombi The Construction

Triangle Trees

  • Bdd. faces of triangulated outerplanar graphs have a binary tree

structure.

Noga Alon and Ohad N. Feldheim Three Edge Lengths Suffice For Drawing Outerplanar Graphs

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SLIDE 26

Preliminaries Results Approaching the Problem Remarks and Open Problems Triangle Trees Naive Approach Embedding Rhombi The Construction

Triangle Trees

  • Bdd. faces of triangulated outerplanar graphs have a binary tree

structure.

Noga Alon and Ohad N. Feldheim Three Edge Lengths Suffice For Drawing Outerplanar Graphs

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SLIDE 27

Preliminaries Results Approaching the Problem Remarks and Open Problems Triangle Trees Naive Approach Embedding Rhombi The Construction

Triangle Trees

  • Bdd. faces of triangulated outerplanar graphs have a binary tree

structure.

Noga Alon and Ohad N. Feldheim Three Edge Lengths Suffice For Drawing Outerplanar Graphs

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Preliminaries Results Approaching the Problem Remarks and Open Problems Triangle Trees Naive Approach Embedding Rhombi The Construction

First Attempt

It would therefore be enough to draw every triangle tree G.

Noga Alon and Ohad N. Feldheim Three Edge Lengths Suffice For Drawing Outerplanar Graphs

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Preliminaries Results Approaching the Problem Remarks and Open Problems Triangle Trees Naive Approach Embedding Rhombi The Construction

First Attempt

It would therefore be enough to draw every triangle tree G. Na¨ ıve attempt: Take a general triangle T. Map every face of G to a copy of T preserving orientation.

Noga Alon and Ohad N. Feldheim Three Edge Lengths Suffice For Drawing Outerplanar Graphs

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Preliminaries Results Approaching the Problem Remarks and Open Problems Triangle Trees Naive Approach Embedding Rhombi The Construction

First Attempt

It would therefore be enough to draw every triangle tree G. Na¨ ıve attempt: Take a general triangle T. Map every face of G to a copy of T preserving orientation. We get a lattice...

Noga Alon and Ohad N. Feldheim Three Edge Lengths Suffice For Drawing Outerplanar Graphs

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SLIDE 31

Preliminaries Results Approaching the Problem Remarks and Open Problems Triangle Trees Naive Approach Embedding Rhombi The Construction

First Attempt

It would therefore be enough to draw every triangle tree G. Na¨ ıve attempt: Take a general triangle T. Map every face of G to a copy of T preserving orientation. We get a lattice...

Noga Alon and Ohad N. Feldheim Three Edge Lengths Suffice For Drawing Outerplanar Graphs

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SLIDE 32

Preliminaries Results Approaching the Problem Remarks and Open Problems Triangle Trees Naive Approach Embedding Rhombi The Construction

First Attempt

It would therefore be enough to draw every triangle tree G. Na¨ ıve attempt: Take a general triangle T. Map every face of G to a copy of T preserving orientation. We get a lattice...

Noga Alon and Ohad N. Feldheim Three Edge Lengths Suffice For Drawing Outerplanar Graphs

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Preliminaries Results Approaching the Problem Remarks and Open Problems Triangle Trees Naive Approach Embedding Rhombi The Construction

First Attempt

It would therefore be enough to draw every triangle tree G. Na¨ ıve attempt: Take a general triangle T. Map every face of G to a copy of T preserving orientation. We get a lattice... ...and fail due to commutativity.

Noga Alon and Ohad N. Feldheim Three Edge Lengths Suffice For Drawing Outerplanar Graphs

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Preliminaries Results Approaching the Problem Remarks and Open Problems Triangle Trees Naive Approach Embedding Rhombi The Construction

Rhombus

Rhombus graph, base edge in red. It is simpler to work with rhombi then with triangles. Every triangle tree is covered by a tree of rhombi.

Noga Alon and Ohad N. Feldheim Three Edge Lengths Suffice For Drawing Outerplanar Graphs

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Preliminaries Results Approaching the Problem Remarks and Open Problems Triangle Trees Naive Approach Embedding Rhombi The Construction

Embedding the Rhombi

External edges are always of length 1. x ∈ T the unit circle. x 1 x + 1

Noga Alon and Ohad N. Feldheim Three Edge Lengths Suffice For Drawing Outerplanar Graphs

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SLIDE 36

Preliminaries Results Approaching the Problem Remarks and Open Problems Triangle Trees Naive Approach Embedding Rhombi The Construction

Embedding the Rhombi

External edges are always of length 1. x ∈ T the unit circle. x 1 x + 1 We embed adjacent rhombi always turning to the left.

Noga Alon and Ohad N. Feldheim Three Edge Lengths Suffice For Drawing Outerplanar Graphs

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SLIDE 37

Preliminaries Results Approaching the Problem Remarks and Open Problems Triangle Trees Naive Approach Embedding Rhombi The Construction

How does it look with one rhombus type

x 2x 2x + x2 2x + x2 + x3

Noga Alon and Ohad N. Feldheim Three Edge Lengths Suffice For Drawing Outerplanar Graphs

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SLIDE 38

Preliminaries Results Approaching the Problem Remarks and Open Problems Triangle Trees Naive Approach Embedding Rhombi The Construction

How does it look with one rhombus type

x 2x 2x + x2 2x + x2 + x3

Noga Alon and Ohad N. Feldheim Three Edge Lengths Suffice For Drawing Outerplanar Graphs

slide-39
SLIDE 39

Preliminaries Results Approaching the Problem Remarks and Open Problems Triangle Trees Naive Approach Embedding Rhombi The Construction

How does it look with one rhombus type

x 2x 2x + x2 2x + x2 + x3

Noga Alon and Ohad N. Feldheim Three Edge Lengths Suffice For Drawing Outerplanar Graphs

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SLIDE 40

Preliminaries Results Approaching the Problem Remarks and Open Problems Triangle Trees Naive Approach Embedding Rhombi The Construction

How does it look with one rhombus type

x 2x 2x + x2 2x + x2 + x3

Noga Alon and Ohad N. Feldheim Three Edge Lengths Suffice For Drawing Outerplanar Graphs

slide-41
SLIDE 41

Preliminaries Results Approaching the Problem Remarks and Open Problems Triangle Trees Naive Approach Embedding Rhombi The Construction

How does it look with one rhombus type

x 2x 2x + x2 2x + x2 + x3

Noga Alon and Ohad N. Feldheim Three Edge Lengths Suffice For Drawing Outerplanar Graphs

slide-42
SLIDE 42

Preliminaries Results Approaching the Problem Remarks and Open Problems Triangle Trees Naive Approach Embedding Rhombi The Construction

How does it look with one rhombus type

x 2x 2x + x2 2x + x2 + x3

Noga Alon and Ohad N. Feldheim Three Edge Lengths Suffice For Drawing Outerplanar Graphs

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SLIDE 43

Preliminaries Results Approaching the Problem Remarks and Open Problems Triangle Trees Naive Approach Embedding Rhombi The Construction

How does it look with one rhombus type

x 2x 2x + x2 2x + x2 + x3 The position of a vertex is thus kixi.

Noga Alon and Ohad N. Feldheim Three Edge Lengths Suffice For Drawing Outerplanar Graphs

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SLIDE 44

Preliminaries Results Approaching the Problem Remarks and Open Problems Triangle Trees Naive Approach Embedding Rhombi The Construction

Embedding the Rhombi - 2

It is enough to show that different vertices are mapped to different polynomials.

Noga Alon and Ohad N. Feldheim Three Edge Lengths Suffice For Drawing Outerplanar Graphs

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SLIDE 45

Preliminaries Results Approaching the Problem Remarks and Open Problems Triangle Trees Naive Approach Embedding Rhombi The Construction

Embedding the Rhombi - 2

It is enough to show that different vertices are mapped to different

  • polynomials. Local obstruction:

Noga Alon and Ohad N. Feldheim Three Edge Lengths Suffice For Drawing Outerplanar Graphs

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SLIDE 46

Preliminaries Results Approaching the Problem Remarks and Open Problems Triangle Trees Naive Approach Embedding Rhombi The Construction

Embedding the Rhombi - 2

It is enough to show that different vertices are mapped to different

  • polynomials. Local obstruction:

Solution: using two kinds of rhombi. The position of a vertex is thus

i

  • j kijxiy j.

Noga Alon and Ohad N. Feldheim Three Edge Lengths Suffice For Drawing Outerplanar Graphs

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SLIDE 47

Preliminaries Results Approaching the Problem Remarks and Open Problems Triangle Trees Naive Approach Embedding Rhombi The Construction

Our Construction

Noga Alon and Ohad N. Feldheim Three Edge Lengths Suffice For Drawing Outerplanar Graphs

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SLIDE 48

Preliminaries Results Approaching the Problem Remarks and Open Problems Remarks Open Problems

What’s not on this talk

What’s not on this talk?

The actual proof

Noga Alon and Ohad N. Feldheim Three Edge Lengths Suffice For Drawing Outerplanar Graphs

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SLIDE 49

Preliminaries Results Approaching the Problem Remarks and Open Problems Remarks Open Problems

What’s not on this talk

What’s not on this talk?

The actual proof Why the construction yields degenerate drawings

Noga Alon and Ohad N. Feldheim Three Edge Lengths Suffice For Drawing Outerplanar Graphs

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SLIDE 50

Preliminaries Results Approaching the Problem Remarks and Open Problems Remarks Open Problems

What’s not on this talk

What’s not on this talk?

The actual proof Why the construction yields degenerate drawings

Noga Alon and Ohad N. Feldheim Three Edge Lengths Suffice For Drawing Outerplanar Graphs

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SLIDE 51

Preliminaries Results Approaching the Problem Remarks and Open Problems Remarks Open Problems

What’s not on this talk

What’s not on this talk?

The actual proof Why the construction yields degenerate drawings How do we think this can be overcome

Noga Alon and Ohad N. Feldheim Three Edge Lengths Suffice For Drawing Outerplanar Graphs

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SLIDE 52

Preliminaries Results Approaching the Problem Remarks and Open Problems Remarks Open Problems

Open Problems

Open Problems

Tighten the bounds. In particular –

Noga Alon and Ohad N. Feldheim Three Edge Lengths Suffice For Drawing Outerplanar Graphs

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SLIDE 53

Preliminaries Results Approaching the Problem Remarks and Open Problems Remarks Open Problems

Open Problems

Open Problems

Tighten the bounds. In particular – Is 2 the degenerate distance number of every outerplanar graph?

Noga Alon and Ohad N. Feldheim Three Edge Lengths Suffice For Drawing Outerplanar Graphs

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SLIDE 54

Preliminaries Results Approaching the Problem Remarks and Open Problems Remarks Open Problems

Open Problems

Open Problems

Tighten the bounds. In particular – Is 2 the degenerate distance number of every outerplanar graph? Except outerplanarity and bdd. degree + bdd. treewidth, what other properties bound the asymptotics of the (degenerate) distance number?

Noga Alon and Ohad N. Feldheim Three Edge Lengths Suffice For Drawing Outerplanar Graphs

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SLIDE 55