Windrose Planarity Embedding Graphs with Direction-Constrained - PowerPoint PPT Presentation

Angular Drawing Angle categories : 0 ◦ , 90 ◦ , 180 ◦ , 270 ◦ , and 360 ◦ Labeled graph ( G , A ): G plane graph, A labeling of angles Angular drawing : end of segments have slopes ≈ ± 1 270 ◦ 90 ◦ 180 ◦ 90 ◦ 90 ◦ 0 ◦ 180 ◦ 90 ◦ 270 ◦ 0 ◦ 90 ◦ 90 ◦ 90 ◦ 180 ◦ 0 ◦ 0 ◦ 180 ◦ 270 ◦ 0 ◦ 360 ◦

Angular Drawing Angle categories : 0 ◦ , 90 ◦ , 180 ◦ , 270 ◦ , and 360 ◦ Labeled graph ( G , A ): G plane graph, A labeling of angles Angular drawing : end of segments have slopes ≈ ± 1 270 ◦ 90 ◦ 180 ◦ 90 ◦ 90 ◦ 0 ◦ 180 ◦ 90 ◦ 0 ◦ 90 ◦ 270 ◦ 90 ◦ 0 ◦ 180 ◦ 90 ◦ 0 ◦ 180 ◦ 270 ◦ 0 ◦ 360 ◦

Angular Drawing Angle categories : 0 ◦ , 90 ◦ , 180 ◦ , 270 ◦ , and 360 ◦ Labeled graph ( G , A ): G plane graph, A labeling of angles Angular drawing : end of segments have slopes ≈ ± 1 ( G , A ) admits angular drawing if: 270 ◦ 90 ◦ 180 ◦ 90 ◦ 90 ◦ 0 ◦ 180 ◦ 90 ◦ 0 ◦ 90 ◦ 270 ◦ 90 ◦ 0 ◦ 180 ◦ 90 ◦ 0 ◦ 180 ◦ 270 ◦ 0 ◦ 360 ◦

Angular Drawing Angle categories : 0 ◦ , 90 ◦ , 180 ◦ , 270 ◦ , and 360 ◦ Labeled graph ( G , A ): G plane graph, A labeling of angles Angular drawing : end of segments have slopes ≈ ± 1 ( G , A ) admits angular drawing if: • Vertex condition : sum of angle cat. at vertex is 360 ◦ 270 ◦ 90 ◦ 180 ◦ 90 ◦ 90 ◦ 0 ◦ 180 ◦ 90 ◦ 0 ◦ 90 ◦ 270 ◦ 90 ◦ 0 ◦ 180 ◦ 90 ◦ 0 ◦ 180 ◦ 270 ◦ 0 ◦ 360 ◦

Angular Drawing Angle categories : 0 ◦ , 90 ◦ , 180 ◦ , 270 ◦ , and 360 ◦ Labeled graph ( G , A ): G plane graph, A labeling of angles Angular drawing : end of segments have slopes ≈ ± 1 ( G , A ) admits angular drawing if: • Vertex condition : sum of angle cat. at vertex is 360 ◦ 270 ◦ 90 ◦ 180 ◦ 90 ◦ 90 ◦ 0 ◦ 180 ◦ 90 ◦ 0 ◦ 90 ◦ 270 ◦ 90 ◦ 0 ◦ 180 ◦ 90 ◦ 0 ◦ 180 ◦ 270 ◦ 0 ◦ 360 ◦

Angular Drawing Angle categories : 0 ◦ , 90 ◦ , 180 ◦ , 270 ◦ , and 360 ◦ Labeled graph ( G , A ): G plane graph, A labeling of angles Angular drawing : end of segments have slopes ≈ ± 1 ( G , A ) admits angular drawing if: • Vertex condition : sum of angle cat. at vertex is 360 ◦ • Cycle condition : sum of angle cat. at (int.) face of length k is k · 180 ◦ − 360 ◦ 270 ◦ 90 ◦ 180 ◦ 90 ◦ 90 ◦ 0 ◦ 180 ◦ 90 ◦ 0 ◦ 90 ◦ 270 ◦ 90 ◦ 0 ◦ 180 ◦ 90 ◦ 0 ◦ 180 ◦ 270 ◦ 0 ◦ 360 ◦

Angular Drawing Angle categories : 0 ◦ , 90 ◦ , 180 ◦ , 270 ◦ , and 360 ◦ Labeled graph ( G , A ): G plane graph, A labeling of angles Angular drawing : end of segments have slopes ≈ ± 1 ( G , A ) admits angular drawing if: • Vertex condition : sum of angle cat. at vertex is 360 ◦ • Cycle condition : sum of angle cat. at (int.) face of length k is k · 180 ◦ − 360 ◦ 270 ◦ 90 ◦ 180 ◦ 90 ◦ 90 ◦ 0 ◦ 180 ◦ 90 ◦ 0 ◦ 90 ◦ 270 ◦ 90 ◦ 0 ◦ 180 ◦ 90 ◦ 0 ◦ 180 ◦ 270 ◦ 0 ◦ 360 ◦

Angular Drawing Angle categories : 0 ◦ , 90 ◦ , 180 ◦ , 270 ◦ , and 360 ◦ Labeled graph ( G , A ): G plane graph, A labeling of angles Angular drawing : end of segments have slopes ≈ ± 1 ⇒ A is angular labeling ( G , A ) admits angular drawing if: • Vertex condition : sum of angle cat. at vertex is 360 ◦ • Cycle condition : sum of angle cat. at (int.) face of length k is k · 180 ◦ − 360 ◦ 270 ◦ 90 ◦ 180 ◦ 90 ◦ 90 ◦ 0 ◦ 180 ◦ 90 ◦ 0 ◦ 90 ◦ 270 ◦ 90 ◦ 0 ◦ 180 ◦ 90 ◦ 0 ◦ 180 ◦ 270 ◦ 0 ◦ 360 ◦

Angular Drawing Angle categories : 0 ◦ , 90 ◦ , 180 ◦ , 270 ◦ , and 360 ◦ Labeled graph ( G , A ): G plane graph, A labeling of angles Angular drawing : end of segments have slopes ≈ ± 1 ⇒ A is angular labeling ( G , A ) admits angular drawing if: • Vertex condition : sum of angle cat. at vertex is 360 ◦ • Cycle condition : sum of angle cat. at (int.) face of length k is k · 180 ◦ − 360 ◦ 270 ◦ 90 ◦ 180 ◦ angular labeling A 90 ◦ 90 ◦ ⇒ unique q-constraints Q A 0 ◦ 180 ◦ 90 ◦ 0 ◦ 90 ◦ 270 ◦ 90 ◦ 0 ◦ 180 ◦ 90 ◦ 0 ◦ 180 ◦ 270 ◦ 0 ◦ 360 ◦

Angular Drawing Angle categories : 0 ◦ , 90 ◦ , 180 ◦ , 270 ◦ , and 360 ◦ Labeled graph ( G , A ): G plane graph, A labeling of angles Angular drawing : end of segments have slopes ≈ ± 1 ⇒ A is angular labeling ( G , A ) admits angular drawing if: • Vertex condition : sum of angle cat. at vertex is 360 ◦ • Cycle condition : sum of angle cat. at (int.) face of length k is k · 180 ◦ − 360 ◦ 270 ◦ 90 ◦ 180 ◦ angular labeling A 90 ◦ 90 ◦ ⇒ unique q-constraints Q A 0 ◦ 180 ◦ 90 ◦ 0 ◦ 90 ◦ 270 ◦ 90 ◦ 0 ◦ 180 ◦ 90 ◦ 0 ◦ 180 ◦ 270 ◦ 0 ◦ 360 ◦

Angular Drawing Angle categories : 0 ◦ , 90 ◦ , 180 ◦ , 270 ◦ , and 360 ◦ Labeled graph ( G , A ): G plane graph, A labeling of angles Angular drawing : end of segments have slopes ≈ ± 1 ⇒ A is angular labeling ( G , A ) admits angular drawing if: • Vertex condition : sum of angle cat. at vertex is 360 ◦ • Cycle condition : sum of angle cat. at (int.) face of length k is k · 180 ◦ − 360 ◦ 270 ◦ 90 ◦ 180 ◦ angular labeling A 90 ◦ 90 ◦ ⇒ unique q-constraints Q A 0 ◦ 180 ◦ 90 ◦ 0 ◦ 90 ◦ 270 ◦ 90 ◦ q-constraints Q 0 ◦ 180 ◦ 90 ◦ 0 ◦ 180 ◦ + large-angle assignment L 270 ◦ ⇒ unique angular labeling A Q , L 0 ◦ 360 ◦

Angular Drawing Angle categories : 0 ◦ , 90 ◦ , 180 ◦ , 270 ◦ , and 360 ◦ Labeled graph ( G , A ): G plane graph, A labeling of angles Angular drawing : end of segments have slopes ≈ ± 1 ⇒ A is angular labeling ( G , A ) admits angular drawing if: • Vertex condition : sum of angle cat. at vertex is 360 ◦ • Cycle condition : sum of angle cat. at (int.) face of length k is k · 180 ◦ − 360 ◦ 270 ◦ 90 ◦ 180 ◦ angular labeling A 90 ◦ 90 ◦ ⇒ unique q-constraints Q A 0 ◦ 180 ◦ 90 ◦ 0 ◦ 90 ◦ 270 ◦ 90 ◦ q-constraints Q 0 ◦ 180 ◦ 90 ◦ 0 ◦ 180 ◦ + large-angle assignment L 270 ◦ angular drawing ⇒ unique angular labeling A Q , L 0 ◦ = windrose planar drawing ˆ 360 ◦

Triangulated Graphs 0 180 0

Triangulated Graphs 0 180 0

Triangulated Graphs 0 90 90

Triangulated Graphs 0 90 90 • No (int.) > 180 ◦ angle categories

Triangulated Graphs 0 90 90 • No (int.) > 180 ◦ angle categories • At least one 0 ◦ angle category per (int.) face

Triangulated Graphs 0 90 90 • No (int.) > 180 ◦ angle categories • At least one 0 ◦ angle category per (int.) face

Triangulated Graphs 0 90 90 • No (int.) > 180 ◦ angle categories • At least one 0 ◦ angle category per (int.) face G ↑

Triangulated Graphs 0 90 90 • No (int.) > 180 ◦ angle categories • At least one 0 ◦ angle category per (int.) face Lemma. Let ( G , A Q ) be a triangulated angular labeled graph. Then, G ↑ is acyclic. G ↑

Triangulated Graphs 0 90 90 • No (int.) > 180 ◦ angle categories • At least one 0 ◦ angle category per (int.) face Lemma. Let ( G , A Q ) be a triangulated angular labeled graph. Then, G ↑ is acyclic and has no internal sources or sinks. G ↑ 270 ◦ 270 ◦

Triangulated Graphs 0 90 90 • No (int.) > 180 ◦ angle categories • At least one 0 ◦ angle category per (int.) face Lemma. Let ( G , A Q ) be a triangulated angular labeled graph. Then, G ↑ is acyclic and has no internal sources or sinks. G → 270 ◦ 270 ◦

Triangulated Graphs 0 90 90 • No (int.) > 180 ◦ angle categories • At least one 0 ◦ angle category per (int.) face Lemma. Let ( G , A Q ) be a triangulated angular labeled graph. Then, G ↑ and G → are acyclic and have no internal sources or sinks. G →

Triangulated Graphs 0 90 90 • No (int.) > 180 ◦ angle categories • At least one 0 ◦ angle category per (int.) face Lemma. Let ( G , A Q ) be a triangulated angular labeled graph. Then, G ↑ and G → are acyclic and have no internal sources or sinks. G

Triangulated Graphs 0 90 90 • No (int.) > 180 ◦ angle categories • At least one 0 ◦ angle category per (int.) face internally Lemma. Let ( G , A Q ) be a triangulated angular labeled graph. Then, G ↑ and G → are acyclic and have no internal sources or sinks. G

Triangulated Graphs 0 90 90 • No (int.) > 180 ◦ angle categories • At least one 0 ◦ angle category per (int.) face internally Lemma. Let ( G , A Q ) be a triangulated angular labeled graph. Then, G ↑ and G → are acyclic and have no internal sources or sinks. G What if there are no (int.) 180 ◦ angle categories?

Quasi-triangulated Graphs What if there are no (int.) 180 ◦ angle categories?

Quasi-triangulated Graphs What if there are no (int.) 180 ◦ angle categories? w N w W w S

Quasi-triangulated Graphs What if there are no (int.) 180 ◦ angle categories? w N w E w W w S

Quasi-triangulated Graphs What if there are no (int.) 180 ◦ angle categories? G → w N w E w W w S

Quasi-triangulated Graphs What if there are no (int.) 180 ◦ angle categories? • topological order on G → : x -coordinates G → w N w E w W w S

Quasi-triangulated Graphs What if there are no (int.) 180 ◦ angle categories? • topological order on G → : x -coordinates G → w N 2 w E 8 14 11 4 6 7 10 12 5 9 3 1 w W 13 w S

Quasi-triangulated Graphs What if there are no (int.) 180 ◦ angle categories? • topological order on G → : x -coordinates G → w N 2 w E 8 14 11 4 6 7 10 12 5 9 3 1 w W 13 w S 1 3 5 7 9 11 13

Quasi-triangulated Graphs What if there are no (int.) 180 ◦ angle categories? • topological order on G → : x -coordinates G ↑ w N w E w W w S 1 3 5 7 9 11 13

Quasi-triangulated Graphs What if there are no (int.) 180 ◦ angle categories? • topological order on G → : x -coordinates • topological order on G ↑ : y -coordinates G ↑ w N w E w W w S 1 3 5 7 9 11 13

Quasi-triangulated Graphs What if there are no (int.) 180 ◦ angle categories? • topological order on G → : x -coordinates • topological order on G ↑ : y -coordinates G ↑ w N 14 w E 13 12 11 10 9 8 7 6 5 4 3 2 w W 1 w S 1 3 5 7 9 11 13

Quasi-triangulated Graphs What if there are no (int.) 180 ◦ angle categories? • topological order on G → : x -coordinates • topological order on G ↑ : y -coordinates G ↑ w N 14 w E 13 12 11 10 9 8 7 6 5 4 3 2 w W 1 w S 1 3 5 7 9 11 13

Quasi-triangulated Graphs What if there are no (int.) 180 ◦ angle categories? • topological order on G → : x -coordinates • topological order on G ↑ : y -coordinates G ↑ w N 14 w E 13 12 11 10 9 8 7 6 5 4 3 2 w W 1 w S 1 3 5 7 9 11 13

Quasi-triangulated Graphs What if there are no (int.) 180 ◦ angle categories? • topological order on G → : x -coordinates • topological order on G ↑ : y -coordinates G ↑ w N 14 w E 13 12 11 10 9 8 7 6 5 4 3 2 w W 1 w S 1 3 5 7 9 11 13

Quasi-triangulated Graphs What if there are no (int.) 180 ◦ angle categories? • topological order on G → : x -coordinates • topological order on G ↑ : y -coordinates G ↑ w N 14 w E 13 12 11 10 9 8 7 6 5 4 3 2 w W 1 w S 1 3 5 7 9 11 13

Quasi-triangulated Graphs What if there are no (int.) 180 ◦ angle categories? • topological order on G → : x -coordinates • topological order on G ↑ : y -coordinates 1 3 5 7 9 11 13

Quasi-triangulated Graphs What if there are no (int.) 180 ◦ angle categories? • topological order on G → : x -coordinates • topological order on G ↑ : y -coordinates 1 3 5 7 9 11 13

Quasi-triangulated Graphs What if there are no (int.) 180 ◦ angle categories? • topological order on G → : x -coordinates • topological order on G ↑ : y -coordinates 0 ◦ 1 3 5 7 9 11 13

Quasi-triangulated Graphs What if there are no (int.) 180 ◦ angle categories? • topological order on G → : x -coordinates • topological order on G ↑ : y -coordinates 90 ◦ 90 ◦ 0 ◦ 1 3 5 7 9 11 13

Quasi-triangulated Graphs What if there are no (int.) 180 ◦ angle categories? • topological order on G → : x -coordinates • topological order on G ↑ : y -coordinates 90 ◦ 90 ◦ 0 ◦ 1 3 5 7 9 11 13

Quasi-triangulated Graphs What if there are no (int.) 180 ◦ angle categories? • topological order on G → : x -coordinates • topological order on G ↑ : y -coordinates 90 ◦ 90 ◦ 0 ◦ 1 3 5 7 9 11 13

Quasi-triangulated Graphs What if there are no (int.) 180 ◦ angle categories? • topological order on G → : x -coordinates • topological order on G ↑ : y -coordinates 90 ◦ 90 ◦ 0 ◦ Lemma. quasi-triangulated angular labeled graph ( G , A Q ), all internal angles have category 0 ◦ or 90 ◦ ⇒ straight-line windrose planar drawing on n × n grid in O ( n ) time 1 3 5 7 9 11 13

Triangulated graphs Let ( G , A Q ) be a triangulated angular labeled graph.

Triangulated graphs Let ( G , A Q ) be a triangulated angular labeled graph. Task: Augment ( G , A Q ) to a quasi-triangulated angular labeled graph ( G ∗ , A Q ∗ ) without internal angle category 180 ◦ .

Triangulated graphs Let ( G , A Q ) be a triangulated angular labeled graph. Task: Augment ( G , A Q ) to a quasi-triangulated angular labeled graph ( G ∗ , A Q ∗ ) without internal angle category 180 ◦ . v

Triangulated graphs Let ( G , A Q ) be a triangulated angular labeled graph. Task: Augment ( G , A Q ) to a quasi-triangulated angular labeled graph ( G ∗ , A Q ∗ ) without internal angle category 180 ◦ . u v