Compact Drawings of 1-Planar Graphs with Right-Angle Crossings and - - PowerPoint PPT Presentation

Compact Drawings of 1-Planar Graphs with Right-Angle Crossings and Few Bends Steven Chaplick, Fabian Lipp, Alexander Wolff, and Johannes Zink

Introduction: Beyond-Planar Graphs

Types of drawings: Planar: No crossings

2

Introduction: Beyond-Planar Graphs

1-Planar: Types of drawings: ≤ 1 crossings per edge K5 Planar: No crossings

2

Introduction: Beyond-Planar Graphs

1-Planar: Types of drawings: ≤ 1 crossings per edge Planar: No crossings

2

Introduction: Beyond-Planar Graphs

1-Planar: Types of drawings: ≤ 1 crossings per edge NIC-Planar: Two crossings share ≤ 1 vertices Planar: No crossings

2

Introduction: Beyond-Planar Graphs

1-Planar: Types of drawings: ≤ 1 crossings per edge NIC-Planar: Two crossings share ≤ 1 vertices Planar: No crossings

2

Introduction: Beyond-Planar Graphs

1-Planar: Types of drawings: ≤ 1 crossings per edge NIC-Planar: Two crossings share ≤ 1 vertices IC-Planar: Two crossings share no vertices Planar: No crossings

2

Introduction: Beyond-Planar Graphs

1-Planar: Types of drawings: ≤ 1 crossings per edge NIC-Planar: Two crossings share ≤ 1 vertices IC-Planar: Two crossings share no vertices Planar: No crossings

2

Introduction: Beyond-Planar Graphs

1-Planar: Types of drawings: ≤ 1 crossings per edge NIC-Planar: Two crossings share ≤ 1 vertices IC-Planar: Two crossings share no vertices RAC: Right angle crossings Planar: No crossings

2

Introduction: Beyond-Planar Graphs

1-Planar: Types of drawings: ≤ 1 crossings per edge NIC-Planar: Two crossings share ≤ 1 vertices IC-Planar: Two crossings share no vertices RAC: Right angle crossings RACk: with ≤ k bends per edge RAC0: with straight-line edges Planar: No crossings

2

Introduction: Beyond-Planar Graphs

1-Planar: Types of drawings: ≤ 1 crossings per edge NIC-Planar: Two crossings share ≤ 1 vertices IC-Planar: Two crossings share no vertices RAC: Right angle crossings RACk: with ≤ k bends per edge RAC0: with straight-line edges poly(n) RACpoly: in polynomial area poly(n) Planar: No crossings                   

• 2

Introduction: Beyond-Planar Graphs

1-Planar: Types of drawings: ≤ 1 crossings per edge NIC-Planar: Two crossings share ≤ 1 vertices IC-Planar: Two crossings share no vertices RAC: Right angle crossings RACk: with ≤ k bends per edge RAC0: with straight-line edges poly(n) RACpoly: in polynomial area poly(n) Planar: No crossings                   

• ⊆

⊆ ⊆

2

Introduction: The Shift Algorithm

[de Fraysseix, Pach, and Pollack, 1990] [Chrobak and Payne, 1995]

3

Introduction: The Shift Algorithm

[de Fraysseix, Pach, and Pollack, 1990] [Chrobak and Payne, 1995] Idea:

3

Introduction: The Shift Algorithm

[de Fraysseix, Pach, and Pollack, 1990] [Chrobak and Payne, 1995] Idea:

• Triangulate given plane graph.

3

Introduction: The Shift Algorithm

[de Fraysseix, Pach, and Pollack, 1990] [Chrobak and Payne, 1995] Idea:

• Triangulate given plane graph.

3

Introduction: The Shift Algorithm

[de Fraysseix, Pach, and Pollack, 1990] [Chrobak and Payne, 1995] Idea:

• Triangulate given plane graph.
• Compute a canonical ordering of

the vertices v1, v2, . . . , vn.

3

Introduction: The Shift Algorithm

[de Fraysseix, Pach, and Pollack, 1990] [Chrobak and Payne, 1995] Idea:

• Triangulate given plane graph.
• Compute a canonical ordering of

the vertices v1, v2, . . . , vn. v6 v5 v4 v3 v1 v2

3

Introduction: The Shift Algorithm

[de Fraysseix, Pach, and Pollack, 1990] [Chrobak and Payne, 1995] Idea:

• Triangulate given plane graph.
• Compute a canonical ordering of

the vertices v1, v2, . . . , vn.

• Draw the graph:

v6 v5 v4 v3 v1 v2

3

Introduction: The Shift Algorithm

[de Fraysseix, Pach, and Pollack, 1990] [Chrobak and Payne, 1995] Idea:

• Triangulate given plane graph.
• Compute a canonical ordering of

the vertices v1, v2, . . . , vn.

• Draw the graph:

– Start with triangle v1, v2, v3. v6 v5 v4 v3 v1 v2

3

Introduction: The Shift Algorithm

[de Fraysseix, Pach, and Pollack, 1990] [Chrobak and Payne, 1995] v1 v2 v3 Idea:

• Triangulate given plane graph.
• Compute a canonical ordering of

the vertices v1, v2, . . . , vn.

• Draw the graph:

– Start with triangle v1, v2, v3. v6 v5 v4 v3 v1 v2

3

Introduction: The Shift Algorithm

[de Fraysseix, Pach, and Pollack, 1990] [Chrobak and Payne, 1995] – For vk: Shift first & last neighbor of vk. v1 v2 v3 Idea:

• Triangulate given plane graph.
• Compute a canonical ordering of

the vertices v1, v2, . . . , vn.

• Draw the graph:

– Start with triangle v1, v2, v3. v6 v5 v4 v3 v1 v2

3

Introduction: The Shift Algorithm

[de Fraysseix, Pach, and Pollack, 1990] [Chrobak and Payne, 1995] – For vk: Shift first & last neighbor of vk. v1 v2 v3 Idea:

• Triangulate given plane graph.
• Compute a canonical ordering of

the vertices v1, v2, . . . , vn.

• Draw the graph:

– Start with triangle v1, v2, v3. v6 v5 v4 v3 v1 v2

3

Introduction: The Shift Algorithm

[de Fraysseix, Pach, and Pollack, 1990] [Chrobak and Payne, 1995] – For vk: Shift first & last neighbor of vk. v1 v2 v3 Idea:

• Triangulate given plane graph.
• Compute a canonical ordering of

the vertices v1, v2, . . . , vn.

• Draw the graph:

– Start with triangle v1, v2, v3. – Add vk to the outer face. v6 v5 v4 v3 v1 v2

3

Introduction: The Shift Algorithm

[de Fraysseix, Pach, and Pollack, 1990] [Chrobak and Payne, 1995] – For vk: Shift first & last neighbor of vk. v1 v2 v3 v4 Idea:

• Triangulate given plane graph.
• Compute a canonical ordering of

the vertices v1, v2, . . . , vn.

• Draw the graph:

– Start with triangle v1, v2, v3. – Add vk to the outer face. v6 v5 v4 v3 v1 v2

3

Introduction: The Shift Algorithm

[de Fraysseix, Pach, and Pollack, 1990] [Chrobak and Payne, 1995] – For vk: Shift first & last neighbor of vk. v1 v2 v3 v4 Idea:

• Triangulate given plane graph.
• Compute a canonical ordering of

the vertices v1, v2, . . . , vn.

• Draw the graph:

– Start with triangle v1, v2, v3. – Add vk to the outer face. v6 v5 v4 v3 v1 v2 ⇒ all slopes on outer face ±1 (except for v1v2)

3

Introduction: The Shift Algorithm

[de Fraysseix, Pach, and Pollack, 1990] [Chrobak and Payne, 1995] – For vk: Shift first & last neighbor of vk. v1 v2 v3 v4 Idea:

• Triangulate given plane graph.
• Compute a canonical ordering of

the vertices v1, v2, . . . , vn.

• Draw the graph:

– Start with triangle v1, v2, v3. – Add vk to the outer face. v6 v5 v4 v3 v1 v2 ⇒ all slopes on outer face ±1 (except for v1v2)

3

Introduction: The Shift Algorithm

[de Fraysseix, Pach, and Pollack, 1990] [Chrobak and Payne, 1995] – For vk: Shift first & last neighbor of vk. v1 v2 v5 v4 v3 Idea:

• Triangulate given plane graph.
• Compute a canonical ordering of

the vertices v1, v2, . . . , vn.

• Draw the graph:

– Start with triangle v1, v2, v3. – Add vk to the outer face. v6 v5 v4 v3 v1 v2 ⇒ all slopes on outer face ±1 (except for v1v2)

3

Introduction: The Shift Algorithm

[de Fraysseix, Pach, and Pollack, 1990] [Chrobak and Payne, 1995] – For vk: Shift first & last neighbor of vk. v4 v5 v1 v2 v3 Idea:

• Triangulate given plane graph.
• Compute a canonical ordering of

the vertices v1, v2, . . . , vn.

• Draw the graph:

– Start with triangle v1, v2, v3. – Add vk to the outer face. v6 v5 v4 v3 v1 v2 ⇒ all slopes on outer face ±1 (except for v1v2)

3

Introduction: The Shift Algorithm

[de Fraysseix, Pach, and Pollack, 1990] [Chrobak and Payne, 1995] – For vk: Shift first & last neighbor of vk. v4 v5 v1 v2 v3 v6 Idea:

• Triangulate given plane graph.
• Compute a canonical ordering of

the vertices v1, v2, . . . , vn.

• Draw the graph:

– Start with triangle v1, v2, v3. – Add vk to the outer face. v6 v5 v4 v3 v1 v2 ⇒ all slopes on outer face ±1 (except for v1v2)

3

Introduction: The Shift Algorithm

[de Fraysseix, Pach, and Pollack, 1990] [Chrobak and Payne, 1995] – For vk: Shift first & last neighbor of vk. v4 v5 v1 v2 v3 v6 Idea:

• Triangulate given plane graph.
• Compute a canonical ordering of

the vertices v1, v2, . . . , vn.

• Draw the graph:

– Start with triangle v1, v2, v3. – Add vk to the outer face. v6 v5 v4 v3 v1 v2

Resulting grid size: (2n − 4) × (n − 2)

⇒ all slopes on outer face ±1 (except for v1v2)

3

Introduction: The Shift Algorithm

[de Fraysseix, Pach, and Pollack, 1990] [Chrobak and Payne, 1995] – For vk: Shift first & last neighbor of vk. v4 v5 v1 v2 v3 v6 Idea:

• Triangulate given plane graph.
• Compute a canonical ordering of

the vertices v1, v2, . . . , vn.

• Draw the graph:

– Start with triangle v1, v2, v3. – Add vk to the outer face. v6 v5 v4 v3 v1 v2

Resulting grid size: (2n − 4) × (n − 2)

⇒ all slopes on outer face ±1 (except for v1v2)

3

Introduction: The Shift Algorithm

[de Fraysseix, Pach, and Pollack, 1990] [Chrobak and Payne, 1995] – For vk: Shift first & last neighbor of vk. v4 v5 v1 v2 v3 v6 Idea:

• Triangulate given plane graph.
• Compute a canonical ordering of

the vertices v1, v2, . . . , vn.

• Draw the graph:

– Start with triangle v1, v2, v3. – Add vk to the outer face. v6 v5 v4 v3 v1 v2

Resulting grid size: (2n − 4) × (n − 2)

⇒ all slopes on outer face ±1 (except for v1v2)

3

Introduction: Related Work

= {graph G | G has a NIC-planar drawing}

planar IC-planar NIC-planar 1-planar

E? contained in (even for fixed embedding) contained in (unknown for fixed embedding) incomparable 4

Introduction: Related Work

planar IC-planar NIC-planar 1-planar RAC1 RAC0 RAC2

E? contained in (even for fixed embedding) contained in (unknown for fixed embedding) incomparable 4

Introduction: Related Work

planar IC-planar NIC-planar 1-planar RACpoly

1

RAC1 RAC0 RACpoly RACpoly

2

RAC2

E? contained in (even for fixed embedding) contained in (unknown for fixed embedding) incomparable

RACpoly

3

4

Introduction: Related Work

[de Fraysseix, Pach, Pollack, 1990] [Schnyder, 1990]

planar IC-planar NIC-planar 1-planar RACpoly

1

RAC1 RAC0 RACpoly RACpoly

2

RAC2

E? contained in (even for fixed embedding) contained in (unknown for fixed embedding) incomparable

RACpoly

3

4

Introduction: Related Work

planar IC-planar NIC-planar 1-planar RACpoly

1

RAC1 RAC0 RACpoly RACpoly

2

RAC2 RACpoly

3

= all graphs E?

[Didimo, Eades, Liotta, 2009] E? contained in (even for fixed embedding) contained in (unknown for fixed embedding) incomparable

E? RACpoly

3

= all graphs

4

Introduction: Related Work

[Didimo, Eades, Liotta, 2009] [Eades, Liotta, 2011]

planar IC-planar NIC-planar 1-planar RACpoly

1

RAC1 RAC0 RACpoly RACpoly

2

RAC2 RACpoly

3

= all graphs E?

E? contained in (even for fixed embedding) contained in (unknown for fixed embedding) incomparable 4

Introduction: Related Work

[Brandenburg, Didimo, Evans, Kindermann, Liotta, Montecchiani, 2015]

planar IC-planar NIC-planar 1-planar RACpoly

1

RAC1 RAC0 RACpoly RACpoly

2

RAC2 RACpoly

3

= all graphs E? E?

E? contained in (even for fixed embedding) contained in (unknown for fixed embedding) incomparable

E?

4

Introduction: Related Work

[Liotta, Montecchiani, 2015]

planar IC-planar NIC-planar 1-planar RACpoly

1

RAC1 RAC0 RACpoly RACpoly

2

RAC2 RACpoly

3

= all graphs E? E?

E? contained in (even for fixed embedding) contained in (unknown for fixed embedding) incomparable 4

Introduction: Related Work

[Didimo, Liotta, Mehrabi, Montecchiani, 2016]

planar IC-planar NIC-planar 1-planar RACpoly

1

RAC1 RAC0 RACpoly RACpoly

2

RAC2 RACpoly

3

= all graphs E? E? E?

E? contained in (even for fixed embedding) contained in (unknown for fixed embedding) incomparable

E?

4

Introduction: Related Work

[Bachmaier, Brandenburg, Hanauer, Neuwirth, Reislhuber, 2017]

planar IC-planar NIC-planar 1-planar RACpoly

1

RAC1 RAC0 RACpoly RACpoly

2

RAC2 RACpoly

3

= all graphs E? E? E?

E? contained in (even for fixed embedding) contained in (unknown for fixed embedding) incomparable 4

Introduction: Related Work

Our results

planar IC-planar NIC-planar 1-planar RACpoly

1

RAC1 RAC0 RACpoly RACpoly

2

RAC2 RACpoly

3

= all graphs E? E? E?

E? contained in (even for fixed embedding) contained in (unknown for fixed embedding) incomparable 4

Introduction: Related Work

Our results

planar IC-planar NIC-planar 1-planar RACpoly

1

RAC1 RAC0 RACpoly RACpoly

2

RAC2 RACpoly

3

= all graphs E? E?

w/o B-configuration

E?

E? contained in (even for fixed embedding) contained in (unknown for fixed embedding) incomparable 4

Introduction: Related Work

planar IC-planar NIC-planar 1-planar RACpoly

1

RAC1 RAC0 RACpoly RACpoly

2

RAC2 RACpoly

3

= all graphs E? E?

w/o B-configuration

E?

Our first main result:

E? contained in (even for fixed embedding) contained in (unknown for fixed embedding) incomparable

NIC-plane graphs ⊆ RACpoly

1

4

Introduction: Related Work

planar IC-planar NIC-planar 1-planar RACpoly

1

RAC1 RAC0 RACpoly RACpoly

2

RAC2 RACpoly

3

= all graphs E? E?

w/o B-configuration

E?

Our first main result:

E? contained in (even for fixed embedding) contained in (unknown for fixed embedding) incomparable

NIC-plane graphs ⊆ RACpoly

1

Our second main result: 1-plane graphs ⊆ RACpoly

2

4

Result 1: NIC-Plane Graphs ⊆ RACpoly

1

planar IC-planar NIC-planar 1-planar RACpoly

1

RAC1 RAC0 RACpoly RACpoly

2

RAC2 RACpoly

3

= all graphs E?

w/o B-configuration

5

Result 1: NIC-Plane Graphs ⊆ RACpoly

1

• Input: a NIC-plane graph

6

Result 1: NIC-Plane Graphs ⊆ RACpoly

1 Approach that nearly works:

• Input: a NIC-plane graph

6

Result 1: NIC-Plane Graphs ⊆ RACpoly

1 Approach that nearly works:

• Input: a NIC-plane graph

6

Result 1: NIC-Plane Graphs ⊆ RACpoly

1

• Enclose each crossing by a so called empty kite:

Approach that nearly works:

• Input: a NIC-plane graph

6

Result 1: NIC-Plane Graphs ⊆ RACpoly

1

• Enclose each crossing by a so called empty kite:

Approach that nearly works:

vdummy

• Input: a NIC-plane graph

6

Result 1: NIC-Plane Graphs ⊆ RACpoly

1

• Replace each pair of crossing edges by a single edge
• Enclose each crossing by a so called empty kite:

Approach that nearly works:

vdummy

• Input: a NIC-plane graph

6

Result 1: NIC-Plane Graphs ⊆ RACpoly

1

• Replace each pair of crossing edges by a single edge
• Enclose each crossing by a so called empty kite:

Approach that nearly works:

vdummy

• Input: a NIC-plane graph

6

Result 1: NIC-Plane Graphs ⊆ RACpoly

1

• Replace each pair of crossing edges by a single edge
• Enclose each crossing by a so called empty kite:
• Draw the obtained plane graph with the Shift Algorithm

Approach that nearly works:

vdummy

• Input: a NIC-plane graph

6

Result 1: NIC-Plane Graphs ⊆ RACpoly

1

• Replace each pair of crossing edges by a single edge
• Enclose each crossing by a so called empty kite:
• Draw the obtained plane graph with the Shift Algorithm

Approach that nearly works:

vdummy

• Input: a NIC-plane graph

6

Result 1: NIC-Plane Graphs ⊆ RACpoly

1

• Replace each pair of crossing edges by a single edge
• Enclose each crossing by a so called empty kite:
• Draw the obtained plane graph with the Shift Algorithm
• Manually reinsert the removed edges with 1 bend so that

they cross in a right angle (crossings and bends on the grid) Approach that nearly works:

vdummy

• Input: a NIC-plane graph

6

Result 1: NIC-Plane Graphs ⊆ RACpoly

1

• Replace each pair of crossing edges by a single edge
• Enclose each crossing by a so called empty kite:
• Draw the obtained plane graph with the Shift Algorithm
• Manually reinsert the removed edges with 1 bend so that

they cross in a right angle (crossings and bends on the grid) Approach that nearly works:

vdummy

• Input: a NIC-plane graph

6

Result 1: NIC-Plane Graphs ⊆ RACpoly

1

• Replace each pair of crossing edges by a single edge
• Enclose each crossing by a so called empty kite:
• Draw the obtained plane graph with the Shift Algorithm
• Manually reinsert the removed edges with 1 bend so that

they cross in a right angle (crossings and bends on the grid) Approach that nearly works:

vdummy

• Input: a NIC-plane graph

6

Result 1: NIC-Plane Graphs ⊆ RACpoly

1

• Replace each pair of crossing edges by a single edge
• Enclose each crossing by a so called empty kite:
• Draw the obtained plane graph with the Shift Algorithm
• Manually reinsert the removed edges with 1 bend so that

they cross in a right angle (crossings and bends on the grid) Approach that nearly works:

vdummy

grid point on the Thales’ circle

• Input: a NIC-plane graph

6

Result 1: NIC-Plane Graphs ⊆ RACpoly

1

• Replace each pair of crossing edges by a single edge
• Enclose each crossing by a so called empty kite:
• Draw the obtained plane graph with the Shift Algorithm
• Manually reinsert the removed edges with 1 bend so that

they cross in a right angle (crossings and bends on the grid) Approach that nearly works:

vdummy

grid point on the Thales’ circle

• Input: a NIC-plane graph

6

Result 1: NIC-Plane Graphs ⊆ RACpoly

1

• Replace each pair of crossing edges by a single edge
• Enclose each crossing by a so called empty kite:
• Draw the obtained plane graph with the Shift Algorithm
• Manually reinsert the removed edges with 1 bend so that

they cross in a right angle (crossings and bends on the grid) Approach that nearly works:

vdummy

grid points for the bends

• Input: a NIC-plane graph

6

Result 1: NIC-Plane Graphs ⊆ RACpoly

1

• Replace each pair of crossing edges by a single edge
• Enclose each crossing by a so called empty kite:
• Draw the obtained plane graph with the Shift Algorithm
• Manually reinsert the removed edges with 1 bend so that

they cross in a right angle (crossings and bends on the grid) Approach that nearly works:

vdummy

• Input: a NIC-plane graph

6

Result 1: NIC-Plane Graphs ⊆ RACpoly

1

• Replace each pair of crossing edges by a single edge
• Enclose each crossing by a so called empty kite:
• Draw the obtained plane graph with the Shift Algorithm
• Manually reinsert the removed edges with 1 bend so that

they cross in a right angle (crossings and bends on the grid) Approach that nearly works:

vdummy

• Input: a NIC-plane graph

6

Result 1: NIC-Plane Graphs ⊆ RACpoly

1

• Replace each pair of crossing edges by a single edge
• Enclose each crossing by a so called empty kite:
• Draw the obtained plane graph with the Shift Algorithm
• Manually reinsert the removed edges with 1 bend so that

they cross in a right angle (crossings and bends on the grid) Approach that nearly works:

vdummy

• Input: a NIC-plane graph

6

Result 1: NIC-Plane Graphs ⊆ RACpoly

1

• Replace each pair of crossing edges by a single edge
• Enclose each crossing by a so called empty kite:
• Draw the obtained plane graph with the Shift Algorithm
• Manually reinsert the removed edges with 1 bend so that

they cross in a right angle (crossings and bends on the grid) Approach that nearly works:

vdummy

• very slim
• Input: a NIC-plane graph

6

Result 1: NIC-Plane Graphs ⊆ RACpoly

1

• Replace each pair of crossing edges by a single edge
• Enclose each crossing by a so called empty kite:
• Draw the obtained plane graph with the Shift Algorithm
• Manually reinsert the removed edges with 1 bend so that

they cross in a right angle (crossings and bends on the grid) Approach that nearly works:

vdummy

• Input: a NIC-plane graph

6

Result 1: NIC-Plane Graphs ⊆ RACpoly

1

• Replace each pair of crossing edges by a single edge
• Enclose each crossing by a so called empty kite:
• Draw the obtained plane graph with the Shift Algorithm
• Manually reinsert the removed edges with 1 bend so that

they cross in a right angle (crossings and bends on the grid) Approach that nearly works:

vdummy

• Input: a NIC-plane graph

6

Result 1: NIC-Plane Graphs ⊆ RACpoly

1

• Replace each pair of crossing edges by a single edge
• Enclose each crossing by a so called empty kite:
• Draw the obtained plane graph with the Shift Algorithm
• Manually reinsert the removed edges with 1 bend so that

they cross in a right angle (crossings and bends on the grid) Approach that nearly works:

vdummy

• Input: a NIC-plane graph

6

Result 1: NIC-Plane Graphs ⊆ RACpoly

1

• Replace each pair of crossing edges by a single edge
• Enclose each crossing by a so called empty kite:
• Draw the obtained plane graph with the Shift Algorithm
• Manually reinsert the removed edges with 1 bend so that

they cross in a right angle (crossings and bends on the grid) Approach that nearly works:

vdummy

• Input: a NIC-plane graph

6

Result 1: NIC-Plane Graphs ⊆ RACpoly

1

• Replace each pair of crossing edges by a single edge
• Enclose each crossing by a so called empty kite:
• Draw the obtained plane graph with the Shift Algorithm
• Manually reinsert the removed edges with 1 bend so that

they cross in a right angle (crossings and bends on the grid) Approach that nearly works:

vdummy

• bad

configu- ration!

• Input: a NIC-plane graph

6

Result 1: NIC-Plane Graphs ⊆ RACpoly

1

bad configu- ration! 7

Result 1: NIC-Plane Graphs ⊆ RACpoly

1 Solution:

bad configu- ration!

• Make the first vertex in the qudrangle

(regarding the canonical ordering) adjacent to the other three vertices.

7

Result 1: NIC-Plane Graphs ⊆ RACpoly

1 Solution:

bad configu- ration!

• Make the first vertex in the qudrangle

(regarding the canonical ordering) adjacent to the other three vertices.

• Use the algorithm by Harel and Sardas

(Shift Algorithm for biconnected graphs).

7

Result 1: NIC-Plane Graphs ⊆ RACpoly

1 Solution:

bad configu- ration!

• Make the first vertex in the qudrangle

(regarding the canonical ordering) adjacent to the other three vertices.

• Use the algorithm by Harel and Sardas

(Shift Algorithm for biconnected graphs). It builds a canonical ordering bottom-up instead of top-down.

7

Result 1: NIC-Plane Graphs ⊆ RACpoly

1 Solution:

bad configu- ration!

• Make the first vertex in the qudrangle

(regarding the canonical ordering) adjacent to the other three vertices.

start with an empty quadrangle

• Use the algorithm by Harel and Sardas

(Shift Algorithm for biconnected graphs). It builds a canonical ordering bottom-up instead of top-down.

7

Result 1: NIC-Plane Graphs ⊆ RACpoly

1 Solution:

bad configu- ration!

• Make the first vertex in the qudrangle

(regarding the canonical ordering) adjacent to the other three vertices. v42

• Use the algorithm by Harel and Sardas

(Shift Algorithm for biconnected graphs). It builds a canonical ordering bottom-up instead of top-down.

7

Result 1: NIC-Plane Graphs ⊆ RACpoly

1 Solution:

bad configu- ration!

• Make the first vertex in the qudrangle

(regarding the canonical ordering) adjacent to the other three vertices. v42

Insert the diagonal

• Use the algorithm by Harel and Sardas

(Shift Algorithm for biconnected graphs). It builds a canonical ordering bottom-up instead of top-down.

7

Result 1: NIC-Plane Graphs ⊆ RACpoly

1 Solution:

bad configu- ration!

• Make the first vertex in the qudrangle

(regarding the canonical ordering) adjacent to the other three vertices. v42

Insert the diagonal

• Use the algorithm by Harel and Sardas

(Shift Algorithm for biconnected graphs). It builds a canonical ordering bottom-up instead of top-down.

7

Result 1: NIC-Plane Graphs ⊆ RACpoly

1 Solution:

bad configu- ration!

• Make the first vertex in the qudrangle

(regarding the canonical ordering) adjacent to the other three vertices. v42 v55

• Use the algorithm by Harel and Sardas

(Shift Algorithm for biconnected graphs). It builds a canonical ordering bottom-up instead of top-down.

7

Result 1: NIC-Plane Graphs ⊆ RACpoly

1 Solution:

bad configu- ration!

• Make the first vertex in the qudrangle

(regarding the canonical ordering) adjacent to the other three vertices. v42 v55 v61

• Use the algorithm by Harel and Sardas

(Shift Algorithm for biconnected graphs). It builds a canonical ordering bottom-up instead of top-down.

7

Result 1: NIC-Plane Graphs ⊆ RACpoly

1 Solution:

bad configu- ration!

• Make the first vertex in the qudrangle

(regarding the canonical ordering) adjacent to the other three vertices. v42 v55 v61 v94

• Use the algorithm by Harel and Sardas

(Shift Algorithm for biconnected graphs). It builds a canonical ordering bottom-up instead of top-down.

7

Result 1: NIC-Plane Graphs ⊆ RACpoly

1 Solution:

bad configu- ration!

• Make the first vertex in the qudrangle

(regarding the canonical ordering) adjacent to the other three vertices. v42 v55 v61 v94

• Use the algorithm by Harel and Sardas

(Shift Algorithm for biconnected graphs).

• Now only three “good” cases can appear:

It builds a canonical ordering bottom-up instead of top-down.

7

Result 1: NIC-Plane Graphs ⊆ RACpoly

1 Solution:

bad configu- ration!

• Make the first vertex in the qudrangle

(regarding the canonical ordering) adjacent to the other three vertices. v42 v55 v61 v94

• Use the algorithm by Harel and Sardas

(Shift Algorithm for biconnected graphs).

• Now only three “good” cases can appear:

Case 1 It builds a canonical ordering bottom-up instead of top-down.

7

Result 1: NIC-Plane Graphs ⊆ RACpoly

1 Solution:

bad configu- ration!

• Make the first vertex in the qudrangle

(regarding the canonical ordering) adjacent to the other three vertices. v42 v55 v61 v94

• Use the algorithm by Harel and Sardas

(Shift Algorithm for biconnected graphs).

• Now only three “good” cases can appear:

Case 1 It builds a canonical ordering bottom-up instead of top-down.

7

Result 1: NIC-Plane Graphs ⊆ RACpoly

1 Solution:

bad configu- ration!

• Make the first vertex in the qudrangle

(regarding the canonical ordering) adjacent to the other three vertices. v42 v55 v61 v94

• Use the algorithm by Harel and Sardas

(Shift Algorithm for biconnected graphs).

• Now only three “good” cases can appear:

Case 1 It builds a canonical ordering bottom-up instead of top-down.

7

Result 1: NIC-Plane Graphs ⊆ RACpoly

1 Solution:

bad configu- ration!

• Make the first vertex in the qudrangle

(regarding the canonical ordering) adjacent to the other three vertices. v42 v55 v61 v94

• Use the algorithm by Harel and Sardas

(Shift Algorithm for biconnected graphs).

• Now only three “good” cases can appear:

Case 1 Case 2 It builds a canonical ordering bottom-up instead of top-down.

7

Result 1: NIC-Plane Graphs ⊆ RACpoly

1 Solution:

bad configu- ration!

• Make the first vertex in the qudrangle

(regarding the canonical ordering) adjacent to the other three vertices. v42 v55 v61 v94

• Use the algorithm by Harel and Sardas

(Shift Algorithm for biconnected graphs).

• Now only three “good” cases can appear:

Case 1 Case 2 It builds a canonical ordering bottom-up instead of top-down.

7

Result 1: NIC-Plane Graphs ⊆ RACpoly

1 Solution:

bad configu- ration!

• Make the first vertex in the qudrangle

(regarding the canonical ordering) adjacent to the other three vertices. v42 v55 v61 v94

• Use the algorithm by Harel and Sardas

(Shift Algorithm for biconnected graphs).

• Now only three “good” cases can appear:

Case 1 Case 2 It builds a canonical ordering bottom-up instead of top-down.

7

Result 1: NIC-Plane Graphs ⊆ RACpoly

1 Solution:

bad configu- ration!

• Make the first vertex in the qudrangle

(regarding the canonical ordering) adjacent to the other three vertices. v42 v55 v61 v94

• Use the algorithm by Harel and Sardas

(Shift Algorithm for biconnected graphs).

• Now only three “good” cases can appear:

Case 1 Case 2 Case 3 It builds a canonical ordering bottom-up instead of top-down.

7

Result 1: NIC-Plane Graphs ⊆ RACpoly

1 Solution:

bad configu- ration!

• Make the first vertex in the qudrangle

(regarding the canonical ordering) adjacent to the other three vertices. v42 v55 v61 v94

• Use the algorithm by Harel and Sardas

(Shift Algorithm for biconnected graphs).

• Now only three “good” cases can appear:

Case 1 Case 2 Case 3 It builds a canonical ordering bottom-up instead of top-down.

7

Result 1: NIC-Plane Graphs ⊆ RACpoly

1 Solution:

bad configu- ration!

• Make the first vertex in the qudrangle

(regarding the canonical ordering) adjacent to the other three vertices. v42 v55 v61 v94

• Use the algorithm by Harel and Sardas

(Shift Algorithm for biconnected graphs).

• Now only three “good” cases can appear:

Case 1 Case 2 Case 3 It builds a canonical ordering bottom-up instead of top-down.

7

Result 1: NIC-Plane Graphs ⊆ RACpoly

1 Full example:

8

Result 1: NIC-Plane Graphs ⊆ RACpoly

1 Full example:

8

Result 2: 1-Plane Graphs ⊆ RACpoly

2

planar IC-planar NIC-planar 1-planar RACpoly

1

RAC1 RAC0 RACpoly RACpoly

2

RAC2 RACpoly

3

= all graphs E?

w/o B-configuration

9

Result 2: 1-Plane Graphs ⊆ RACpoly

2

• Input: a 1-plane graph

10

Result 2: 1-Plane Graphs ⊆ RACpoly

2

• Input: a 1-plane graph

Preprocessing:

10

Result 2: 1-Plane Graphs ⊆ RACpoly

2

• Input: a 1-plane graph

Preprocessing:

10

Result 2: 1-Plane Graphs ⊆ RACpoly

2

• Enclose each crossing by a so called subdivided kite:
• Input: a 1-plane graph

Preprocessing:

10

Result 2: 1-Plane Graphs ⊆ RACpoly

2

• Enclose each crossing by a so called subdivided kite:
• Input: a 1-plane graph

Preprocessing:

10

Result 2: 1-Plane Graphs ⊆ RACpoly

2

• Planarize the graph by replacing each crossing by a vertex
• Enclose each crossing by a so called subdivided kite:
• Input: a 1-plane graph

Preprocessing:

10

Result 2: 1-Plane Graphs ⊆ RACpoly

2

• Planarize the graph by replacing each crossing by a vertex
• Enclose each crossing by a so called subdivided kite:
• Input: a 1-plane graph

Preprocessing: c

10

Result 2: 1-Plane Graphs ⊆ RACpoly

2

• Planarize the graph by replacing each crossing by a vertex
• Enclose each crossing by a so called subdivided kite:
• Draw the obtained plane graph

using the Shift Algorithm

• Input: a 1-plane graph

Preprocessing: Drawing phase: c

10

Result 2: 1-Plane Graphs ⊆ RACpoly

2

• Planarize the graph by replacing each crossing by a vertex
• Enclose each crossing by a so called subdivided kite:
• Draw the obtained plane graph

using the Shift Algorithm

• Input: a 1-plane graph

Preprocessing: Drawing phase: c c

10

Result 2: 1-Plane Graphs ⊆ RACpoly

2 c

11

Result 2: 1-Plane Graphs ⊆ RACpoly

2 Postprocessing (obtaining crossings at right angles): c

11

Result 2: 1-Plane Graphs ⊆ RACpoly

2 Postprocessing (obtaining crossings at right angles):

• Consider the four axis-parallel half-lines originating at c

c

11

Result 2: 1-Plane Graphs ⊆ RACpoly

2 Postprocessing (obtaining crossings at right angles):

• Consider the four axis-parallel half-lines originating at c

c

11

Result 2: 1-Plane Graphs ⊆ RACpoly

2

• Assign the four edges being incident to c to these half-lines

Postprocessing (obtaining crossings at right angles):

• Consider the four axis-parallel half-lines originating at c

c

11

Result 2: 1-Plane Graphs ⊆ RACpoly

2

• Assign the four edges being incident to c to these half-lines

Postprocessing (obtaining crossings at right angles):

• Consider the four axis-parallel half-lines originating at c

c

11

Result 2: 1-Plane Graphs ⊆ RACpoly

2

• Bend these edges at their assigned half-lines:
• Assign the four edges being incident to c to these half-lines

Postprocessing (obtaining crossings at right angles):

• Consider the four axis-parallel half-lines originating at c

c

11

Result 2: 1-Plane Graphs ⊆ RACpoly

2

• Bend these edges at their assigned half-lines:
• Assign the four edges being incident to c to these half-lines

Postprocessing (obtaining crossings at right angles):

• Consider the four axis-parallel half-lines originating at c

c

11

Result 2: 1-Plane Graphs ⊆ RACpoly

2

• Bend these edges at their assigned half-lines:
• Assign the four edges being incident to c to these half-lines

Postprocessing (obtaining crossings at right angles):

• Consider the four axis-parallel half-lines originating at c

c

11

Result 2: 1-Plane Graphs ⊆ RACpoly

2

• Bend these edges at their assigned half-lines:
• Assign the four edges being incident to c to these half-lines

Postprocessing (obtaining crossings at right angles):

• Consider the four axis-parallel half-lines originating at c

c Be careful: One assignment might depend on another one

11

Result 2: 1-Plane Graphs ⊆ RACpoly

2

• Bend these edges at their assigned half-lines:
• Assign the four edges being incident to c to these half-lines

Postprocessing (obtaining crossings at right angles):

• Consider the four axis-parallel half-lines originating at c

c Be careful: One assignment might depend on another one Solution: re-draw the independent

• nes first

1 2 1 1

11

Result 2: 1-Plane Graphs ⊆ RACpoly

2

• Bend these edges at their assigned half-lines:
• Assign the four edges being incident to c to these half-lines

Postprocessing (obtaining crossings at right angles):

• Consider the four axis-parallel half-lines originating at c

c Be careful: One assignment might depend on another one Solution: re-draw the independent

• nes first

1 2 1 1 Be careful: There might be no grid points to bend the edges

11

Result 2: 1-Plane Graphs ⊆ RACpoly

2

• Bend these edges at their assigned half-lines:
• Assign the four edges being incident to c to these half-lines

Postprocessing (obtaining crossings at right angles):

• Consider the four axis-parallel half-lines originating at c

c Be careful: One assignment might depend on another one Solution: re-draw the independent

• nes first

1 2 1 1 Be careful: There might be no grid points to bend the edges Solution: make the grid sufficiently fine

11

Result 2: 1-Plane Graphs ⊆ RACpoly

2

• Bend these edges at their assigned half-lines:
• Assign the four edges being incident to c to these half-lines

Postprocessing (obtaining crossings at right angles):

• Consider the four axis-parallel half-lines originating at c

c Be careful: One assignment might depend on another one Solution: re-draw the independent

• nes first

1 2 1 1 Be careful: There might be no grid points to bend the edges Solution: make the grid sufficiently fine

grid size: O(n) × O(n)

11

Result 2: 1-Plane Graphs ⊆ RACpoly

2

• Bend these edges at their assigned half-lines:
• Assign the four edges being incident to c to these half-lines

Postprocessing (obtaining crossings at right angles):

• Consider the four axis-parallel half-lines originating at c

c

• 1. Refine the grid by ˜

n ∈ O(n)

• 2. Re-draw independent edges
• 3. Refine the grid by ˜

n again

• 4. Re-draw dependent edges

grid size: O(n) × O(n)

11

Result 2: 1-Plane Graphs ⊆ RACpoly

2

• Bend these edges at their assigned half-lines:
• Assign the four edges being incident to c to these half-lines

Postprocessing (obtaining crossings at right angles):

• Consider the four axis-parallel half-lines originating at c

c

• 1. Refine the grid by ˜

n ∈ O(n)

• 2. Re-draw independent edges
• 3. Refine the grid by ˜

n again

• 4. Re-draw dependent edges

grid size: O(n2) × O(n2)

11

Result 2: 1-Plane Graphs ⊆ RACpoly

2

• Bend these edges at their assigned half-lines:
• Assign the four edges being incident to c to these half-lines

Postprocessing (obtaining crossings at right angles):

• Consider the four axis-parallel half-lines originating at c

c

• 1. Refine the grid by ˜

n ∈ O(n)

• 2. Re-draw independent edges
• 3. Refine the grid by ˜

n again

• 4. Re-draw dependent edges

grid size: O(n2) × O(n2)

11

Result 2: 1-Plane Graphs ⊆ RACpoly

2

• Bend these edges at their assigned half-lines:
• Assign the four edges being incident to c to these half-lines

Postprocessing (obtaining crossings at right angles):

• Consider the four axis-parallel half-lines originating at c

c

• 1. Refine the grid by ˜

n ∈ O(n)

• 2. Re-draw independent edges
• 3. Refine the grid by ˜

n again

• 4. Re-draw dependent edges

grid size: O(n2) × O(n2)

11

Result 2: 1-Plane Graphs ⊆ RACpoly

2

• Bend these edges at their assigned half-lines:
• Assign the four edges being incident to c to these half-lines

Postprocessing (obtaining crossings at right angles):

• Consider the four axis-parallel half-lines originating at c

c

• 1. Refine the grid by ˜

n ∈ O(n)

• 2. Re-draw independent edges
• 3. Refine the grid by ˜

n again

• 4. Re-draw dependent edges

grid size: O(n2) × O(n2)

11

Result 2: 1-Plane Graphs ⊆ RACpoly

2

• Bend these edges at their assigned half-lines:
• Assign the four edges being incident to c to these half-lines

Postprocessing (obtaining crossings at right angles):

• Consider the four axis-parallel half-lines originating at c

c

• 1. Refine the grid by ˜

n ∈ O(n)

• 2. Re-draw independent edges
• 3. Refine the grid by ˜

n again

• 4. Re-draw dependent edges

grid size: O(n3) × O(n3)

11

Result 2: 1-Plane Graphs ⊆ RACpoly

2

• Bend these edges at their assigned half-lines:
• Assign the four edges being incident to c to these half-lines

Postprocessing (obtaining crossings at right angles):

• Consider the four axis-parallel half-lines originating at c

c

• 1. Refine the grid by ˜

n ∈ O(n)

• 2. Re-draw independent edges
• 3. Refine the grid by ˜

n again

• 4. Re-draw dependent edges

grid size: O(n3) × O(n3)

11

Result 2: 1-Plane Graphs ⊆ RACpoly

2

• Bend these edges at their assigned half-lines:
• Assign the four edges being incident to c to these half-lines

Postprocessing (obtaining crossings at right angles):

• Consider the four axis-parallel half-lines originating at c

c

• 1. Refine the grid by ˜

n ∈ O(n)

• 2. Re-draw independent edges
• 3. Refine the grid by ˜

n again

• 4. Re-draw dependent edges

grid size: O(n3) × O(n3)

11

Result 2: 1-Plane Graphs ⊆ RACpoly

2

• Bend these edges at their assigned half-lines:
• Assign the four edges being incident to c to these half-lines

Postprocessing (obtaining crossings at right angles):

• Consider the four axis-parallel half-lines originating at c

c

• 1. Refine the grid by ˜

n ∈ O(n)

• 2. Re-draw independent edges
• 3. Refine the grid by ˜

n again

• 4. Re-draw dependent edges
• Remove the dummy objects

grid size: O(n3) × O(n3)

11

Result 2: 1-Plane Graphs ⊆ RACpoly

2

• Bend these edges at their assigned half-lines:
• Assign the four edges being incident to c to these half-lines

Postprocessing (obtaining crossings at right angles):

• Consider the four axis-parallel half-lines originating at c
• 1. Refine the grid by ˜

n ∈ O(n)

• 2. Re-draw independent edges
• 3. Refine the grid by ˜

n again

• 4. Re-draw dependent edges
• Remove the dummy objects

grid size: O(n3) × O(n3)

11

Summary and Open Questions

planar IC-planar NIC-planar 1-planar RAC1 RAC0 RACpoly RACpoly

2

RAC2 RACpoly

3

= all graphs E?

w/o B-configuration

NIC-plane ⊆ RACpoly

1

1-plane ⊆ RACpoly

2

Preserves embedding Yes Yes Runtime O(n) O(n) Bends per edge ≤ 1 ≤ 2 Grid size O(n) × O(n) O(n3) × O(n3)

RACpoly

1

12

Summary and Open Questions

planar IC-planar NIC-planar 1-planar RAC1 RAC0 RACpoly RACpoly

2

RAC2 RACpoly

3

= all graphs E?

w/o B-configuration

NIC-plane ⊆ RACpoly

1

1-plane ⊆ RACpoly

2

Preserves embedding Yes Yes Runtime O(n) O(n) Bends per edge ≤ 1 ≤ 2 Grid size O(n) × O(n) O(n3) × O(n3)

RACpoly

1

12

Summary and Open Questions

planar IC-planar NIC-planar 1-planar RAC1 RAC0 RACpoly RACpoly

2

RAC2 RACpoly

3

= all graphs E?

w/o B-configuration

NIC-plane ⊆ RACpoly

1

1-plane ⊆ RACpoly

2

Preserves embedding Yes Yes Runtime O(n) O(n) Bends per edge ≤ 1 ≤ 2 Grid size O(n) × O(n) O(n3) × O(n3)

RACpoly

1

12

Summary and Open Questions

planar IC-planar NIC-planar 1-planar RAC1 RAC0 RACpoly RACpoly

2

RAC2 RACpoly

3

= all graphs E?

w/o B-configuration

NIC-plane ⊆ RACpoly

1

1-plane ⊆ RACpoly

2

Preserves embedding Yes Yes Runtime O(n) O(n) Bends per edge ≤ 1 ≤ 2 Grid size O(n) × O(n) O(n3) × O(n3)

RACpoly

1

12

Summary and Open Questions

planar IC-planar NIC-planar 1-planar RAC1 RAC0 RACpoly RACpoly

2

RAC2 RACpoly

3

= all graphs E?

w/o B-configuration

NIC-plane ⊆ RACpoly

1

1-plane ⊆ RACpoly

2

Preserves embedding Yes Yes Runtime O(n) O(n) Bends per edge ≤ 1 ≤ 2 Grid size O(n) × O(n) O(n3) × O(n3)

RACpoly

1

12

Summary and Open Questions

planar IC-planar NIC-planar 1-planar RAC1 RAC0 RACpoly RACpoly

2

RAC2 RACpoly

3

= all graphs E?

w/o B-configuration

NIC-plane ⊆ RACpoly

1

1-plane ⊆ RACpoly

2

Preserves embedding Yes Yes Runtime O(n) O(n) Bends per edge ≤ 1 ≤ 2 Grid size O(n) × O(n) O(n3) × O(n3)

RACpoly

1

12

Summary and Open Questions

planar IC-planar NIC-planar 1-planar RAC1 RAC0 RACpoly RACpoly

2

RAC2 RACpoly

3

= all graphs E?

w/o B-configuration

NIC-plane ⊆ RACpoly

1

1-plane ⊆ RACpoly

2

Preserves embedding Yes Yes Runtime O(n) O(n) Bends per edge ≤ 1 ≤ 2 Grid size O(n) × O(n) O(n3) × O(n3)

RACpoly

1

12

Summary and Open Questions

Open question: 1-planar ⊆ RACpoly

1

?

planar IC-planar NIC-planar 1-planar RAC1 RAC0 RACpoly RACpoly

2

RAC2 RACpoly

3

= all graphs E?

w/o B-configuration

?

NIC-plane ⊆ RACpoly

1

1-plane ⊆ RACpoly

2

Preserves embedding Yes Yes Runtime O(n) O(n) Bends per edge ≤ 1 ≤ 2 Grid size O(n) × O(n) O(n3) × O(n3)

RACpoly

1

12

Summary and Open Questions

Open question: 1-planar ⊆ RACpoly

1

?

planar IC-planar NIC-planar 1-planar RAC1 RAC0 RACpoly RACpoly

2

RAC2 RACpoly

3

= all graphs ? ? ?

w/o B-configuration

?

More open questions

E?

NIC-plane ⊆ RACpoly

1

1-plane ⊆ RACpoly

2

Preserves embedding Yes Yes Runtime O(n) O(n) Bends per edge ≤ 1 ≤ 2 Grid size O(n) × O(n) O(n3) × O(n3)

RACpoly

1

12

Summary and Open Questions

Open question: 1-planar ⊆ RACpoly

1

?

planar IC-planar NIC-planar 1-planar RAC1 RAC0 RACpoly RACpoly

2

RAC2 RACpoly

3

= all graphs

w/o B-configuration

More open questions NIC-plane ⊆ RACpoly

1

1-plane ⊆ RACpoly

2

Preserves embedding Yes Yes Runtime O(n) O(n) Bends per edge ≤ 1 ≤ 2 Grid size O(n) × O(n) O(n3) × O(n3)

RACpoly

1

? ? ? E? ?

12