1-bend RAC Drawings of 1-Planar Graphs Walter Didimo, Giuseppe - - PowerPoint PPT Presentation

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1-bend RAC Drawings of 1-Planar Graphs Walter Didimo, Giuseppe - - PowerPoint PPT Presentation

Jul 07, 2023 483 likes •1.17k views

1-bend RAC Drawings of 1-Planar Graphs Walter Didimo, Giuseppe Liotta, Saeed Mehrabi, Fabrizio Montecchiani Things to avoid in graph drawing Things to avoid in graph drawing Too many crossings Things to avoid in graph drawing Too many

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SLIDE 1

1-bend RAC Drawings

  • f

1-Planar Graphs

Walter Didimo, Giuseppe Liotta, Saeed Mehrabi, Fabrizio Montecchiani

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SLIDE 2

Things to avoid in graph drawing

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SLIDE 3

Things to avoid in graph drawing

  • Too many crossings
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SLIDE 4

Things to avoid in graph drawing

  • Too many crossings
  • Too many bends
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SLIDE 5

A good property

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SLIDE 6

A good property

  • Right angle crossings (RAC)!

[Huang, Hong, Eades – 2008]

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SLIDE 7

A good property

1-bend 1-planar RAC drawing

[Huang, Hong, Eades – 2008]

  • Right angle crossings (RAC)!
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SLIDE 8

Questions

  • General: Which kind of graphs can be drawn with:
  • a few crossings per edge,
  • a few bends per edge,
  • right angle crossings (RAC)?
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SLIDE 9

Questions

  • General: Which kind of graphs can be drawn with:
  • a few crossings per edge,
  • a few bends per edge,
  • right angle crossings (RAC)?
  • Specific: Does every 1-planar graph admit a 1-bend RAC

drawing?

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SLIDE 10

1-planar RAC drawings

  • Not all 1-planar graphs have a straight-line RAC drawing

[consequence of edge density results]

  • Not all straight-line RAC drawable graphs are 1-planar

[Eades and Liotta - 2013]

  • Every 1-plane kite-triangulation has a 1-bend RAC drawing

[Angelini et al. - 2009]

  • Every 1-plane graph with independent crossings (IC-planar) has a

straight-line RAC drawing [Brandenburg et al. - 2013]

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SLIDE 11

Our result

  • Theorem. Every simple 1-planar graph admits a 1-bend RAC

drawing, which can be computed in linear time if an initial 1-planar embedding is given

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SLIDE 12

Some definitions

1-plane graph (not necessarily simple)

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Some definitions

1-plane graph (not necessarily simple) kite

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Some definitions

1-plane graph (not necessarily simple) empty kite

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Some definitions

1-plane graph (not necessarily simple) not a kite!

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SLIDE 16

Observation

triangulated 1-plane graph (not necessarily simple)

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Observation

triangulated 1-plane graph (not necessarily simple) Every pair of crossing edges forms an empty kyte except possibly for a pair of crossing edges on the outer face empty kite

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Observation

triangulated 1-plane graph (not necessarily simple) Every pair of crossing edges forms an empty kyte except possibly for a pair of crossing edges on the outer face not a kite

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SLIDE 19

Algorithm Outline

G simple 1-plane G+ triangulated 1-plane (multi-edges) G* hierarchical contraction of G+  1-bend 1-planar RAC drawing

  • f G

+ 1-bend 1-planar RAC drawing of G+

augmentation (the embedding may change) recursive procedure recursive procedure removal of dummy elements input put

  • utpu

put

1 2 3 4

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SLIDE 20

Algorithm Outline

G simple 1-plane G+ triangulated 1-plane (multi-edges) G* hierarchical contraction of G+  1-bend 1-planar RAC drawing

  • f G

+ 1-bend 1-planar RAC drawing of G+

augmentation (the embedding may change) recursive procedure recursive procedure removal of dummy elements input put

  • utpu

put

1 2 3 4

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SLIDE 21

Augmentation

G simple 1-plane

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SLIDE 22

Augmentation

G simple 1-plane for each pair ir of cross

  • ssin

ing g edges ges add an enclosing 4-cycle

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SLIDE 23

Augmentation

G simple 1-plane for each pair ir of cross

  • ssin

ing g edges ges add an enclosing 4-cycle

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SLIDE 24

Augmentation

G simple 1-plane for each pair ir of cross

  • ssin

ing g edges ges add an enclosing 4-cycle

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SLIDE 25

Augmentation

G simple 1-plane for each pair ir of cross

  • ssin

ing g edges ges add an enclosing 4-cycle

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SLIDE 26

Augmentation

G simple 1-plane remove those multiple edges that belong to the input graph

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SLIDE 27

Augmentation

G simple 1-plane

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SLIDE 28

Augmentation

G simple 1-plane remove one (multiple) edge from each face of degree two, if any

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SLIDE 29

Augmentation

G simple 1-plane tria iang ngulat ulate faces of degree > 3 by inserting a star inside them

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Augmentation

G+ triangulated 1-plane

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SLIDE 31

Algorithm Outline

G simple 1-plane G+ triangulated 1-plane (multi-edges) G* hierarchical contraction of G+  1-bend 1-planar RAC drawing

  • f G

+ 1-bend 1-planar RAC drawing of G+

augmentation (the embedding may change) recursive procedure recursive procedure removal of dummy elements input put

  • utpu

put

1 2 3 4

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SLIDE 32

Property of G+

G+ triangulated 1-plane

  • triangular faces
  • multiple edges

never crossed

  • nly empty kites
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SLIDE 33

Property of G+

G+ triangulated 1-plane

  • triangular faces
  • multiple edges

never crossed

  • nly empty kites

structure of each separation pair

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SLIDE 34

Property of G+

G+ triangulated 1-plane

  • triangular faces
  • multiple edges

never crossed

  • nly empty kites

structure of each separation pair

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SLIDE 35

Hierarchical contraction

G+ triangulated 1-plane contract all inner components of each separation pair into a thic ick k edge ge structure of each separation pair

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Hierarchical contraction

G+ triangulated 1-plane contraction contract all inner components of each separation pair into a thic ick k edge ge

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SLIDE 37

Hierarchical contraction

G+ triangulated 1-plane contraction contract all inner components of each separation pair into a thic ick k edge ge

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SLIDE 38

Hierarchical contraction

G+ triangulated 1-plane

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Hierarchical contraction

G+ triangulated 1-plane

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Hierarchical contraction

G+ triangulated 1-plane G* hierarchical contraction

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Hierarchical contraction

G+ triangulated 1-plane G* hierarchical contraction simple 3-connected triangulated 1-plane graph

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Algorithm Outline

G simple 1-plane G+ triangulated 1-plane (multi-edges) G* hierarchical contraction of G+  1-bend 1-planar RAC drawing

  • f G

+ 1-bend 1-planar RAC drawing of G+

augmentation (the embedding may change) recursive procedure recursive procedure removal of dummy elements input put

  • utpu

put

1 2 3 4

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SLIDE 43

Drawing procedure

apply Chiba et

  • al. 1984

convex faces and prescribed outerface

remove crossing edges

3-connected plane graph

reinsert crossing edges

partial drawing

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Drawing procedure

partial drawing

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Drawing procedure

partial drawing

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Drawing procedure

partial drawing

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Drawing procedure

partial drawing

remove crossing edges

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Drawing procedure

partial drawing

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SLIDE 49

Drawing procedure

partial drawing

apply Chiba et

  • al. 1984
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Drawing procedure

partial drawing

reinsert crossing edges

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SLIDE 51

Drawing procedure

partial drawing

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SLIDE 52

Drawing procedure

partial drawing

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Drawing procedure

partial drawing

remove crossing edges

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SLIDE 54

Drawing procedure

partial drawing

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SLIDE 55

Drawing procedure

partial drawing

apply Chiba et

  • al. 1984
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SLIDE 56

Drawing procedure

partial drawing

reinsert crossing edges

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SLIDE 57

Drawing procedure

new partial drawing

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SLIDE 58

Drawing procedure

new partial drawing

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SLIDE 59

Drawing procedure

new partial drawing

draw it as usual

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SLIDE 60

Drawing procedure

+ 1-bend 1-planar RAC drawing of G+

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Algorithm Outline

G simple 1-plane G+ triangulated 1-plane (multi-edges) G* hierarchical contraction of G+  1-bend 1-planar RAC drawing

  • f G

+ 1-bend 1-planar RAC drawing of G+

augmentation (the embedding may change) recursive procedure recursive procedure removal of dummy elements input put

  • utpu

put

1 2 3 4

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SLIDE 62

Drawing procedure

+ 1-bend 1-planar RAC drawing of G+

remove dummy elements

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SLIDE 63

Drawing procedure

 1-bend 1-planar RAC drawing of G input graph G

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Drawing procedure

input graph G

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Open problems

  • Our algorithm may give rise to drawings with exponential area:

is such an area necessary in some cases?

  • Our algorithm is allowed to change the initial embedding:

What if we cannot?

  • Still missing:

Characterization of straight-line 1-planar RAC graphs

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SLIDE 66

Thank you