Graphs with large total angular resolution Oswin Aichholzer Matias - - PowerPoint PPT Presentation

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Graphs with large total angular resolution Oswin Aichholzer Matias - - PowerPoint PPT Presentation

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W I S S E N T E C H N I K L E I D E N S C H A F T Graphs with large total angular resolution Oswin Aichholzer Matias Korman Yoshio Okamoto Irene Parada Daniel Perz Andr e van Renssen Birgit Vogtenhuber 18.

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W I S S E N T E C H N I K L E I D E N S C H A F T www.tugraz.at

Graphs with large total angular resolution

Oswin Aichholzer Matias Korman Yoshio Okamoto Irene Parada Daniel Perz Andr´ e van Renssen Birgit Vogtenhuber

  • 18. September 2019
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Introduction

Definition

Definition (Total angular resolution)

The total angular resolution of a straight-line drawing is the minimum angle between two intersecting edges of the drawing. The total angular resolution of a graph G, or short TAR(G), is the maximum total angular resolution over all straight-line drawings of this graph.

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Introduction

Motivation

Crossing resolution Angular resolution Total angular resolution

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Introduction

Considered questions

Can we find an upper bound for the number of edges

  • f graphs G with TAR(G) > 60◦?

What is the complexity of deciding whether TAR(G) ≥ 60◦?

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5

Introduction

Upper bounds for the number of edges

Number of edges of drawings with:

crossing resolution 90◦: ≤ 4n − 10 [Didimo, Eades, Liotta, 2011] crossing resolution greater than 60◦: ≤ 6.5n − 10 [Ackermann, Tardos, 2007] total angular resolution greater than 60◦: ≤ 2n − 6 with some small exceptions [This work]

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Upper bound for the number of edges

Planarized drawing

Planarized drawing: replace every crossing by a vertex.

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Upper bound for the number of edges

Planarized drawing

Planarized drawing: replace every crossing by a vertex.

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Upper bound for the number of edges

Size of a cell

Size of a cell: number of sides in planarized drawing incident to this cell.

C

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Upper bound for the number of edges

Size of a cell

Size of a cell: number of sides in planarized drawing incident to this cell.

C Size of cell C: 4

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Upper bound for the number of edges

Size of a cell

Size of a cell: number of sides in planarized drawing incident to this cell.

C Size of cell C: 6

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Upper bound for the number of edges

Basic idea

Let D be a drawing. If TAR(D) > 60◦, then D does not contain a triangle and no three edges cross in one point. So every cell has at least size 4.

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Upper bound for the number of edges

Basic idea

Let D be a drawing. If TAR(D) > 60◦, then D does not contain a triangle and no three edges cross in one point. So every cell has at least size 4. Lemma

Given a connected drawing D with n ≥ 1 vertices and m

  • edges. The unbounded cell of D has size k and

TAR(D) > 60◦. Then m ≤ 2n − 2 − ⌈k/2⌉.

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Upper bound for the number of edges

m ≤ 2n − 4

Lemma

Given a drawing D with TAR(D) > 60◦. If the unbound cell has size at least 4, then m ≤ 2n − 4.

The only possible triangle-free drawings with an unbound cell of size at most 2 are:

the empty graph a single vertex two vertices joined by an edge.

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Upper bound for the number of edges

Idea to continue

D’

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Upper bound for the number of edges

Idea to continue

D’

m′ ≤ 2n′ − 4 m′ ≥ m − 8 n′ = n − 5 m ≤ 2n − 6

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Upper bound for the number of edges

Exceptions

E0 E1 E2 E4 E5 E3 E6 E7 E8 E9

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Upper bound for the number of edges

Result

Theorem

Given a graph G with TAR(G) > 60◦. Then m ≤ 2n − 6 or G is in the exceptions.

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Upper bound for the number of edges

Tightness

Drawing of a graph with TAR(G) > 60◦ and 2n − 6 edges.

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Upper bound for the number of edges

Tightness

Drawing of a graph with TAR(G) > 60◦ and 2n − 6 edges.

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NP-hardness for TAR(G) ≥ 60◦

Hardness results

Before: It is NP-hard to decide whether a graph G has angular resolution ≥ 90◦. [Forman et al. 1993]

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NP-hardness for TAR(G) ≥ 60◦

Hardness results

Before: It is NP-hard to decide whether a graph G has total angular resolution ≥ 90◦. [Forman et al. 1993]

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NP-hardness for TAR(G) ≥ 60◦

Hardness results

Before: It is NP-hard to decide whether a graph G has total angular resolution ≥ 90◦. [Forman et al. 1993] Theorem

It is NP-hard to decide whether a graph G has TAR(G) ≥ 60◦.

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NP-hardness for TAR(G) ≥ 60◦

Hardness results

Before: It is NP-hard to decide whether a graph G has total angular resolution ≥ 90◦. [Forman et al. 1993] Theorem

It is NP-hard to decide whether a graph G has TAR(G) ≥ 60◦.

Proof by reduction from 3SAT.

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NP-hardness for TAR(G) ≥ 60◦

Construction

ℓ2 ℓ1

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NP-hardness for TAR(G) ≥ 60◦

Variable gadgets

ℓ2 ℓ1 X1

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NP-hardness for TAR(G) ≥ 60◦

Variable gadgets

ℓ2 ℓ1 X1 x1,1 x1,2 x1,3 x1,4 x1,1 x1,2 x1,3 x1,4

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NP-hardness for TAR(G) ≥ 60◦

Variable gadgets

ℓ2 ℓ1 x1,1 x1,2 x1,3 x1,4 x1,1 x1,2 x1,3 x1,4 X1

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NP-hardness for TAR(G) ≥ 60◦

Clause gadget

C1 C2 C3 C4 ℓ2 ℓ1 X1 x1,1 x1,2 x1,3 x1,4 x1,1 x1,2 x1,3 x1,4

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NP-hardness for TAR(G) ≥ 60◦

Clause gadget

C1 ℓ2 ℓ1 X1 x1,1 x1,2 x1,3 x1,4 x1,1 x1,2 x1,3 x1,4

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NP-hardness for TAR(G) ≥ 60◦

Connections

Cj Xi,j Xi,j Cj Xi,j Xi,j Cj Xi,j Xi,j left side of variable gadget right side of variable gadget Connection to:

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NP-hardness for TAR(G) ≥ 60◦

Connections

C1 ℓ2 ℓ1 not possible with ≥ 60◦

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NP-hardness for TAR(G) ≥ 60◦

Connections

C1 ℓ2 ℓ1 not possible with ≥ 60◦

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NP-hardness for TAR(G) ≥ 60◦

Example

C1 C2 C3 C4 ℓ2 ℓ1 X1 X2 X3 (x1 ∨ x2 ∨ x3) ∧ (x1 ∨ x2 ∨ x3) ∧ (x1 ∨ x2 ∨ x3) ∧ (x1 ∨ x2 ∨ x3)

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Open problems

Open problems

Do almost all graphs with TAR(G) > k−2

k 90◦ have at most 2n−2−⌊k 2⌋ edges?

At which angle(s) α does the decision problem, whether TAR(G) ≥ α, change from NP-hard to polynomially solvable?

  • 18. September 2019