Straight line drawing of a graph on the flat torus Luca Castelli - PowerPoint PPT Presentation

Straight line drawing of a graph on the flat torus Luca Castelli Aleardi, LIX Olivier Devillers, INRIA ´ Eric Fusy, LIX 1

Torus 2

Torus 2

Torus Flat torus 2

Torus Flat torus 2

Torus Flat torus 2

Torus Flat torus Graph 2

Problem statement Torus Flat torus Graph Convex straight line drawing 2

Result Given a map on a torus 3

Result Given a map on a torus (essentially 3-connected) Get a Convex straight line drawing 3 2 ) in rectangle O ( n ) × O ( n 3

[De Fraysseix, Pach, & Pollack] Given a triangulation Get a Planar straight line drawing in rectangle O ( n ) × O ( n ) 4

Algorithm, global view triangulation of a cylinder, no chords, no 2-cycles Vertices ordering Incremental drawing (boudary characteristics) triangulation of a cylinder, with chords, 2-cycles & loops Split triangulation in pieces maps of a cylinder Different boudary characteristics triangulation and maps of a torus Cut the torus → cylinder 5

Algorithm, vertices ordering triangulation of a cylinder, no chords, no 2-cycles k j h i e a b 6

Algorithm, vertices ordering triangulation of a cylinder, no chords, no 2-cycles Annular view / periodic view k j h h i i j e h k i e f b a e g c f d a a b a b g c d k j lower boundary 6

Algorithm, vertices ordering triangulation of a cylinder, no chords, no 2-cycles Annular view / periodic view h i j e h k i f b a e g c f d a a b g c d k j lower boundary no chords at lower boundary 6

Algorithm, vertices ordering triangulation of a cylinder, no chords, no 2-cycles Annular view / periodic view h i j e h k i f b a e g c f d a a b g c d k j lower boundary no 2-cycles 6

Algorithm, vertices ordering triangulation of a cylinder, no chords, no 2-cycles Shelling order 6

Algorithm, vertices ordering triangulation of a cylinder, no chords, no 2-cycles Remove vertices, so that the remaining part is an annulus 7 6

Algorithm, vertices ordering triangulation of a cylinder, no chords, no 2-cycles 6 7 6

Algorithm, vertices ordering triangulation of a cylinder, no chords, no 2-cycles 6 5 7 6

Algorithm, vertices ordering triangulation of a cylinder, no chords, no 2-cycles 6 4 5 7 6

Algorithm, vertices ordering triangulation of a cylinder, no chords, no 2-cycles 6 4 3 5 7 6

Algorithm, vertices ordering triangulation of a cylinder, no chords, no 2-cycles 6 4 3 2 5 7 6

Algorithm, vertices ordering triangulation of a cylinder, no chords, no 2-cycles 6 4 3 2 5 7 6

Algorithm, incremental drawing triangulation of a cylinder, no chords, no 2-cycles incremental drawing 7

Algorithm, incremental drawing triangulation of a cylinder, no chords, no 2-cycles incremental drawing slopes +1 or -1 7

Algorithm, incremental drawing triangulation of a cylinder, no chords, no 2-cycles incremental drawing slopes +1 or -1 7

Algorithm, incremental drawing triangulation of a cylinder, no chords, no 2-cycles incremental drawing slopes +1 or -1 7

Algorithm, incremental drawing triangulation of a cylinder, no chords, no 2-cycles incremental drawing slopes +1 or -1 7

Algorithm, incremental drawing triangulation of a cylinder, no chords, no 2-cycles incremental drawing 7

Algorithm, incremental drawing triangulation of a cylinder, no chords, no 2-cycles incremental drawing | slopes | < 1 7

Algorithm, incremental drawing triangulation of a cylinder, no chords, no 2-cycles incremental drawing | slopes | < 1 7

Algorithm, incremental drawing triangulation of a cylinder, no chords, no 2-cycles 8

Algorithm, incremental drawing triangulation of a cylinder, no chords, no 2-cycles 8

Algorithm, incremental drawing triangulation of a cylinder, no chords, no 2-cycles 1 8

Algorithm, incremental drawing triangulation of a cylinder, no chords, no 2-cycles 2 1 8

Algorithm, incremental drawing triangulation of a cylinder, no chords, no 2-cycles 3 2 1 8

Algorithm, incremental drawing triangulation of a cylinder, no chords, no 2-cycles 4 3 2 1 8

Algorithm, incremental drawing triangulation of a cylinder, no chords, no 2-cycles 5 4 3 2 1 8

Algorithm, incremental drawing triangulation of a cylinder, no chords, no 2-cycles 5 6 4 3 2 1 8

Algorithm, incremental drawing triangulation of a cylinder, no chords, no 2-cycles 7 5 6 4 3 2 1 8

Algorithm, incremental drawing triangulation of a cylinder, no chords, no 2-cycles 7 5 6 4 3 2 1 height ≤ 2n 8

Algorithm, incremental drawing triangulation of a cylinder, no chords, no 2-cycles 7 5 6 4 height ≤ 2n 3 2 1 height ≤ 2n 8

Algorithm, incremental drawing triangulation of a cylinder, no chords, no 2-cycles 7 height ≤ 2n 5 6 4 height ≤ 2n 3 2 1 height ≤ 2n 8

Algorithm, incremental drawing triangulation of a cylinder, no chords, no 2-cycles width ≤ 2 n 7 5 height ≤ 6 4 2 n × ( c + 1) 3 2 1 8

Algorithm, 2-cycles and loops 9

Algorithm, 2-cycles and loops 9

Algorithm, 2-cycles and loops loop 2-cycle 9

Algorithm, 2-cycles and loops 9

Algorithm, 2-cycles and loops 9

Algorithm, 2-cycles and loops 9

Algorithm, 2-cycles and loops 9

Algorithm, chords 10

Algorithm, chords 10

Algorithm, chords chords 10

Algorithm, chords 10

Algorithm, chords 10

Algorithm, maps several adaptations slopes +1 or -1 or 0 11

Algorithm, from cylinder to torus 12

Algorithm, from cylinder to torus 12

Algorithm, from cylinder to torus tambourine 12

Algorithm, from cylinder to torus cylinder 12

Algorithm, from cylinder to torus cylinder 12

Algorithm, from cylinder to torus torus 12

Given a map on a torus 13

Given a map on a torus (essentially 3-connected) Get a Convex straight line drawing on grid 2 n × (1 + 2 n ( c + 1)) c ≤ √ 2 n 13