1-Bend RAC Drawings of NIC-Planar Graphs in Quadratic Area Steven - - PowerPoint PPT Presentation

1-Bend RAC Drawings

• f NIC-Planar Graphs

in Quadratic Area

Steven Chaplick, Fabian Lipp, Alexander Wolff, and Johannes Zink

Beyond-Planar Graphs

Types of Drawings: Planar: No crossings

2

Beyond-Planar Graphs

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

2

Beyond-Planar Graphs

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

2

Beyond-Planar Graphs

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

2

Beyond-Planar Graphs

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

2

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

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

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

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 RAC1: with ≤ 1 bends per edge RAC0: with straight-line edges Planar: No crossings

2

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 RAC1: with ≤ 1 bends per edge RAC0: with straight-line edges poly(n) RACpoly: in polynomial area poly(n) Planar: No crossings                   

• 2

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 RAC1: with ≤ 1 bends per edge RAC0: with straight-line edges poly(n) RACpoly: in polynomial area poly(n) Planar: No crossings                   

• 2

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 3

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 3

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

3

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

3

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

3

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 3

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?

3

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 3

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?

3

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 3

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 3

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 3

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 3

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 3

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 main result:

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

NIC-plane graphs ⊆ RACpoly

1

3

NIC-plane graphs ⊆ RACpoly

1

4

NIC-plane graphs ⊆ RACpoly

1 Algorithm in O(n) time:

4

NIC-plane graphs ⊆ RACpoly

1 Algorithm in O(n) time: Input: NIC-plane graph (G, E) with n vertices

4

NIC-plane graphs ⊆ RACpoly

1 Algorithm in O(n) time: Graph G with a NIC-planar embedding E Input: NIC-plane graph (G, E) with n vertices

4

NIC-plane graphs ⊆ RACpoly

1 Algorithm in O(n) time: Graph G with a NIC-planar embedding E Input: NIC-plane graph (G, E) with n vertices Output: 1-bend RAC drawing Γ of G according to E Every vertex, bend point, and crossing point of Γ lies on a grid

• f size O(n) × O(n)

4

The Shift Algorithm

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

5

The Shift Algorithm

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

5

The Shift Algorithm

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

• Triangulate given plane graph.

5

The Shift Algorithm

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

• Triangulate given plane graph.

5

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.

5

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

5

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

5

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

5

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

5

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

5

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

5

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

5

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

5

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)

5

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)

5

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)

5

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)

5

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)

5

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)

5

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)

5

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)

5

Approach that Nearly Works

6

Approach that Nearly Works

• Input: a NIC-plane graph

6

Approach that Nearly Works

• Input: a NIC-plane graph

6

Approach that Nearly Works

• Enclose each crossing by a so called empty kite (

)

• Input: a NIC-plane graph

6

Approach that Nearly Works

• Enclose each crossing by a so called empty kite (

)

• Input: a NIC-plane graph

vdummy

6

Approach that Nearly Works

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

)

• Input: a NIC-plane graph

vdummy

6

Approach that Nearly Works

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

)

• Input: a NIC-plane graph

vdummy

6

Approach that Nearly Works

• 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
• Input: a NIC-plane graph

vdummy

6

Approach that Nearly Works

• 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
• Input: a NIC-plane graph

vdummy

6

Approach that Nearly Works

• 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)

• Input: a NIC-plane graph

vdummy

6

Approach that Nearly Works

• 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)

• Input: a NIC-plane graph

vdummy

6

Approach that Nearly Works

• 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)

• Input: a NIC-plane graph

vdummy

6

Approach that Nearly Works

• 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)

• Input: a NIC-plane graph

vdummy

grid point on the Thales’ circle

6

Approach that Nearly Works

• 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)

• Input: a NIC-plane graph

vdummy

grid point on the Thales’ circle

6

Approach that Nearly Works

• 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)

• Input: a NIC-plane graph

vdummy

grid points for the bends

6

Approach that Nearly Works

• 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)

• Input: a NIC-plane graph

vdummy

• 6

Approach that Nearly Works

• 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)

• Input: a NIC-plane graph

vdummy

• 6

Approach that Nearly Works

• 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)

• Input: a NIC-plane graph

vdummy

• 6

Approach that Nearly Works

• 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)

• Input: a NIC-plane graph

vdummy

• very slim

6

Approach that Nearly Works

• 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)

• Input: a NIC-plane graph

vdummy

• 6

Approach that Nearly Works

• 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)

• Input: a NIC-plane graph

vdummy

• 6

Approach that Nearly Works

• 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)

• Input: a NIC-plane graph

vdummy

• 6

Approach that Nearly Works

• 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)

• Input: a NIC-plane graph

vdummy

• 6

Approach that Nearly Works

• 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)

• Input: a NIC-plane graph

vdummy

• bad

configu- ration! 6

Our Algorithm

bad configu- ration! 7

Our Algorithm

Solution:

bad configu- ration!

• Make the first vertex in the qudrangle

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

7

Our Algorithm

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

Our Algorithm

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

Our Algorithm

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

Our Algorithm

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

Our Algorithm

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

Our Algorithm

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

Our Algorithm

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

Our Algorithm

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

Our Algorithm

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

Our Algorithm

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

Our Algorithm

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

Our Algorithm

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

Our Algorithm

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

Our Algorithm

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

Our Algorithm

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

Our Algorithm

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

Our Algorithm

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

Our Algorithm

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

Our Algorithm

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

Summary

8

Summary

• Runs in O(n) time.

8

Summary

• Runs in O(n) time.
• Resulting drawing is

NIC-planar RAC with ≤ 1 bend per edge.

8

Summary

• Runs in O(n) time.
• Resulting drawing is

NIC-planar RAC with ≤ 1 bend per edge.

• Grid of size at most

(16n − 32) × (8n − 16).

8

Summary

• Runs in O(n) time.
• Resulting drawing is

NIC-planar RAC with ≤ 1 bend per edge.

• Grid of size at most

(16n − 32) × (8n − 16).

• Needs NIC-planar

embedding as input; this embedding is preserved.

8

Summary

• Runs in O(n) time.
• Resulting drawing is

NIC-planar RAC with ≤ 1 bend per edge.

• Grid of size at most

(16n − 32) × (8n − 16).

• Needs NIC-planar

embedding as input; this embedding is preserved.

planar IC-planar NIC-planar 1-planar RACpoly

1

RAC1 RAC0 RACpoly RACpoly

2

RAC2 RACpoly

3

= all graphs E ?

w/o B-configuration

8

Summary

• Runs in O(n) time.
• Resulting drawing is

NIC-planar RAC with ≤ 1 bend per edge.

• Grid of size at most

(16n − 32) × (8n − 16).

• Needs NIC-planar

embedding as input; this embedding is preserved. Our main result: 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

8

Summary

• Runs in O(n) time.
• Resulting drawing is

NIC-planar RAC with ≤ 1 bend per edge.

• Grid of size at most

(16n − 32) × (8n − 16).

• Needs NIC-planar

embedding as input; this embedding is preserved. Open question: 1-planar 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

?

8

Summary

• Runs in O(n) time.
• Resulting drawing is

NIC-planar RAC with ≤ 1 bend per edge.

• Grid of size at most

(16n − 32) × (8n − 16).

• Needs NIC-planar

embedding as input; this embedding is preserved. Open question: 1-planar 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

?

More open questions

E ?

8