windrose planarity - pub.informatik.uni-wuerzburg.de filewindrose planarity embedding graphs with...
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![Page 1: Windrose Planarity - pub.informatik.uni-wuerzburg.de fileWindrose Planarity Embedding Graphs with Direction-Constrained Edges Published at SODA'16. Joint work with Patrizio Angelini,](https://reader031.vdocuments.site/reader031/viewer/2022022807/5cde865488c993680f8d29bb/html5/thumbnails/1.jpg)
Windrose Planarity
Embedding Graphs withDirection-Constrained Edges
Published at SODA’16. Joint work withPatrizio Angelini, Giordano Da Lozzo, Giuseppe Di Battista,
Valentino Di Donato, Gunter Rote & Ignaz Rutter
Dr. Philipp KindermannLG Theoretische Informatik
FernUniversitat in Hagen
![Page 2: Windrose Planarity - pub.informatik.uni-wuerzburg.de fileWindrose Planarity Embedding Graphs with Direction-Constrained Edges Published at SODA'16. Joint work with Patrizio Angelini,](https://reader031.vdocuments.site/reader031/viewer/2022022807/5cde865488c993680f8d29bb/html5/thumbnails/2.jpg)
Upward Planarity
An undirected graph is planar : no crossings
![Page 3: Windrose Planarity - pub.informatik.uni-wuerzburg.de fileWindrose Planarity Embedding Graphs with Direction-Constrained Edges Published at SODA'16. Joint work with Patrizio Angelini,](https://reader031.vdocuments.site/reader031/viewer/2022022807/5cde865488c993680f8d29bb/html5/thumbnails/3.jpg)
Upward Planarity
An undirected graph is planar : no crossings
![Page 4: Windrose Planarity - pub.informatik.uni-wuerzburg.de fileWindrose Planarity Embedding Graphs with Direction-Constrained Edges Published at SODA'16. Joint work with Patrizio Angelini,](https://reader031.vdocuments.site/reader031/viewer/2022022807/5cde865488c993680f8d29bb/html5/thumbnails/4.jpg)
Upward Planarity
An undirected graph is planar : no crossings
A directed graph is upwards planar :
• all edges are y -monotone curves directed upwards
• no crossings
![Page 5: Windrose Planarity - pub.informatik.uni-wuerzburg.de fileWindrose Planarity Embedding Graphs with Direction-Constrained Edges Published at SODA'16. Joint work with Patrizio Angelini,](https://reader031.vdocuments.site/reader031/viewer/2022022807/5cde865488c993680f8d29bb/html5/thumbnails/5.jpg)
Upward Planarity
An undirected graph is planar : no crossings
A directed graph is upwards planar :
• all edges are y -monotone curves directed upwards
• no crossings
![Page 6: Windrose Planarity - pub.informatik.uni-wuerzburg.de fileWindrose Planarity Embedding Graphs with Direction-Constrained Edges Published at SODA'16. Joint work with Patrizio Angelini,](https://reader031.vdocuments.site/reader031/viewer/2022022807/5cde865488c993680f8d29bb/html5/thumbnails/6.jpg)
Upward Planarity
An undirected graph is planar : no crossings
A directed graph is upwards planar :
• all edges are y -monotone curves directed upwards
• no crossings
![Page 7: Windrose Planarity - pub.informatik.uni-wuerzburg.de fileWindrose Planarity Embedding Graphs with Direction-Constrained Edges Published at SODA'16. Joint work with Patrizio Angelini,](https://reader031.vdocuments.site/reader031/viewer/2022022807/5cde865488c993680f8d29bb/html5/thumbnails/7.jpg)
Upward Planarity
planar
An undirected graph is planar : no crossings
A directed graph is upwards planar :
• all edges are y -monotone curves directed upwards
• no crossings
![Page 8: Windrose Planarity - pub.informatik.uni-wuerzburg.de fileWindrose Planarity Embedding Graphs with Direction-Constrained Edges Published at SODA'16. Joint work with Patrizio Angelini,](https://reader031.vdocuments.site/reader031/viewer/2022022807/5cde865488c993680f8d29bb/html5/thumbnails/8.jpg)
Upward Planarity
planaracyclic
An undirected graph is planar : no crossings
A directed graph is upwards planar :
• all edges are y -monotone curves directed upwards
• no crossings
![Page 9: Windrose Planarity - pub.informatik.uni-wuerzburg.de fileWindrose Planarity Embedding Graphs with Direction-Constrained Edges Published at SODA'16. Joint work with Patrizio Angelini,](https://reader031.vdocuments.site/reader031/viewer/2022022807/5cde865488c993680f8d29bb/html5/thumbnails/9.jpg)
Upward Planarity
planaracyclic
An undirected graph is planar : no crossings
A directed graph is upwards planar :
• all edges are y -monotone curves directed upwards
• no crossings
![Page 10: Windrose Planarity - pub.informatik.uni-wuerzburg.de fileWindrose Planarity Embedding Graphs with Direction-Constrained Edges Published at SODA'16. Joint work with Patrizio Angelini,](https://reader031.vdocuments.site/reader031/viewer/2022022807/5cde865488c993680f8d29bb/html5/thumbnails/10.jpg)
Upward Planarity
planaracyclic?
An undirected graph is planar : no crossings
A directed graph is upwards planar :
• all edges are y -monotone curves directed upwards
• no crossings
![Page 11: Windrose Planarity - pub.informatik.uni-wuerzburg.de fileWindrose Planarity Embedding Graphs with Direction-Constrained Edges Published at SODA'16. Joint work with Patrizio Angelini,](https://reader031.vdocuments.site/reader031/viewer/2022022807/5cde865488c993680f8d29bb/html5/thumbnails/11.jpg)
Upward Planarity: Testing
Testing Upward Planarity is...
![Page 12: Windrose Planarity - pub.informatik.uni-wuerzburg.de fileWindrose Planarity Embedding Graphs with Direction-Constrained Edges Published at SODA'16. Joint work with Patrizio Angelini,](https://reader031.vdocuments.site/reader031/viewer/2022022807/5cde865488c993680f8d29bb/html5/thumbnails/12.jpg)
Upward Planarity: Testing
Testing Upward Planarity is...• NP-complete in general [Garg & Tamassia ’95]
![Page 13: Windrose Planarity - pub.informatik.uni-wuerzburg.de fileWindrose Planarity Embedding Graphs with Direction-Constrained Edges Published at SODA'16. Joint work with Patrizio Angelini,](https://reader031.vdocuments.site/reader031/viewer/2022022807/5cde865488c993680f8d29bb/html5/thumbnails/13.jpg)
Upward Planarity: Testing
Testing Upward Planarity is...• NP-complete in general [Garg & Tamassia ’95]• poly for single-source graphs [Di Battista et al. ’98]
![Page 14: Windrose Planarity - pub.informatik.uni-wuerzburg.de fileWindrose Planarity Embedding Graphs with Direction-Constrained Edges Published at SODA'16. Joint work with Patrizio Angelini,](https://reader031.vdocuments.site/reader031/viewer/2022022807/5cde865488c993680f8d29bb/html5/thumbnails/14.jpg)
Upward Planarity: Testing
Testing Upward Planarity is...• NP-complete in general [Garg & Tamassia ’95]• poly for single-source graphs [Di Battista et al. ’98]• poly for fixed embedding [Garg & Tamassia ’95]
![Page 15: Windrose Planarity - pub.informatik.uni-wuerzburg.de fileWindrose Planarity Embedding Graphs with Direction-Constrained Edges Published at SODA'16. Joint work with Patrizio Angelini,](https://reader031.vdocuments.site/reader031/viewer/2022022807/5cde865488c993680f8d29bb/html5/thumbnails/15.jpg)
Upward Planarity: Testing
planaracyclic
Testing Upward Planarity is...• NP-complete in general [Garg & Tamassia ’95]• poly for single-source graphs [Di Battista et al. ’98]• poly for fixed embedding [Garg & Tamassia ’95]
?
![Page 16: Windrose Planarity - pub.informatik.uni-wuerzburg.de fileWindrose Planarity Embedding Graphs with Direction-Constrained Edges Published at SODA'16. Joint work with Patrizio Angelini,](https://reader031.vdocuments.site/reader031/viewer/2022022807/5cde865488c993680f8d29bb/html5/thumbnails/16.jpg)
Upward Planarity: Testing
planaracyclicbimodal
Testing Upward Planarity is...• NP-complete in general [Garg & Tamassia ’95]• poly for single-source graphs [Di Battista et al. ’98]• poly for fixed embedding [Garg & Tamassia ’95]
![Page 17: Windrose Planarity - pub.informatik.uni-wuerzburg.de fileWindrose Planarity Embedding Graphs with Direction-Constrained Edges Published at SODA'16. Joint work with Patrizio Angelini,](https://reader031.vdocuments.site/reader031/viewer/2022022807/5cde865488c993680f8d29bb/html5/thumbnails/17.jpg)
Upward Planarity: Testing
planaracyclicbimodal
Testing Upward Planarity is...• NP-complete in general [Garg & Tamassia ’95]• poly for single-source graphs [Di Battista et al. ’98]• poly for fixed embedding [Garg & Tamassia ’95]
X
![Page 18: Windrose Planarity - pub.informatik.uni-wuerzburg.de fileWindrose Planarity Embedding Graphs with Direction-Constrained Edges Published at SODA'16. Joint work with Patrizio Angelini,](https://reader031.vdocuments.site/reader031/viewer/2022022807/5cde865488c993680f8d29bb/html5/thumbnails/18.jpg)
Upward Planarity: Testing
planaracyclicbimodal
Testing Upward Planarity is...• NP-complete in general [Garg & Tamassia ’95]• poly for single-source graphs [Di Battista et al. ’98]• poly for fixed embedding [Garg & Tamassia ’95]
X ×
![Page 19: Windrose Planarity - pub.informatik.uni-wuerzburg.de fileWindrose Planarity Embedding Graphs with Direction-Constrained Edges Published at SODA'16. Joint work with Patrizio Angelini,](https://reader031.vdocuments.site/reader031/viewer/2022022807/5cde865488c993680f8d29bb/html5/thumbnails/19.jpg)
Windrose Planarity
q-constrained graph (G ,Q):
![Page 20: Windrose Planarity - pub.informatik.uni-wuerzburg.de fileWindrose Planarity Embedding Graphs with Direction-Constrained Edges Published at SODA'16. Joint work with Patrizio Angelini,](https://reader031.vdocuments.site/reader031/viewer/2022022807/5cde865488c993680f8d29bb/html5/thumbnails/20.jpg)
Windrose Planarity
q-constrained graph (G ,Q):• G : undirected planar graph
![Page 21: Windrose Planarity - pub.informatik.uni-wuerzburg.de fileWindrose Planarity Embedding Graphs with Direction-Constrained Edges Published at SODA'16. Joint work with Patrizio Angelini,](https://reader031.vdocuments.site/reader031/viewer/2022022807/5cde865488c993680f8d29bb/html5/thumbnails/21.jpg)
Windrose Planarity
q-constrained graph (G ,Q):• G : undirected planar graph• Q: partition of all neighbors of v into
↗
v ,↖
v ,↙
v , and↘
v .
![Page 22: Windrose Planarity - pub.informatik.uni-wuerzburg.de fileWindrose Planarity Embedding Graphs with Direction-Constrained Edges Published at SODA'16. Joint work with Patrizio Angelini,](https://reader031.vdocuments.site/reader031/viewer/2022022807/5cde865488c993680f8d29bb/html5/thumbnails/22.jpg)
Windrose Planarity
q-constrained graph (G ,Q):• G : undirected planar graph• Q: partition of all neighbors of v into
↗
v ,↖
v ,↙
v , and↘
v .
↗
v↖
v
↘
v↙
v
v
![Page 23: Windrose Planarity - pub.informatik.uni-wuerzburg.de fileWindrose Planarity Embedding Graphs with Direction-Constrained Edges Published at SODA'16. Joint work with Patrizio Angelini,](https://reader031.vdocuments.site/reader031/viewer/2022022807/5cde865488c993680f8d29bb/html5/thumbnails/23.jpg)
Windrose Planarity
Two directions:
q-constrained graph (G ,Q):• G : undirected planar graph• Q: partition of all neighbors of v into
↗
v ,↖
v ,↙
v , and↘
v .
↗
v↖
v
↘
v↙
v
v
![Page 24: Windrose Planarity - pub.informatik.uni-wuerzburg.de fileWindrose Planarity Embedding Graphs with Direction-Constrained Edges Published at SODA'16. Joint work with Patrizio Angelini,](https://reader031.vdocuments.site/reader031/viewer/2022022807/5cde865488c993680f8d29bb/html5/thumbnails/24.jpg)
Windrose Planarity
Two directions:
q-constrained graph (G ,Q):• G : undirected planar graph• Q: partition of all neighbors of v into
↗
v ,↖
v ,↙
v , and↘
v .
↗
v↖
v
↘
v↙
v
A q-constrained graph is windrose planar :
v
![Page 25: Windrose Planarity - pub.informatik.uni-wuerzburg.de fileWindrose Planarity Embedding Graphs with Direction-Constrained Edges Published at SODA'16. Joint work with Patrizio Angelini,](https://reader031.vdocuments.site/reader031/viewer/2022022807/5cde865488c993680f8d29bb/html5/thumbnails/25.jpg)
Windrose Planarity
Two directions:
q-constrained graph (G ,Q):• G : undirected planar graph• Q: partition of all neighbors of v into
↗
v ,↖
v ,↙
v , and↘
v .
↗
v↖
v
↘
v↙
v
A q-constrained graph is windrose planar :• no crossings
v
![Page 26: Windrose Planarity - pub.informatik.uni-wuerzburg.de fileWindrose Planarity Embedding Graphs with Direction-Constrained Edges Published at SODA'16. Joint work with Patrizio Angelini,](https://reader031.vdocuments.site/reader031/viewer/2022022807/5cde865488c993680f8d29bb/html5/thumbnails/26.jpg)
Windrose Planarity
Two directions:
q-constrained graph (G ,Q):• G : undirected planar graph• Q: partition of all neighbors of v into
↗
v ,↖
v ,↙
v , and↘
v .
↗
v↖
v
↘
v↙
v
A q-constrained graph is windrose planar :• no crossings• all edges are xy -monotone curves
v
![Page 27: Windrose Planarity - pub.informatik.uni-wuerzburg.de fileWindrose Planarity Embedding Graphs with Direction-Constrained Edges Published at SODA'16. Joint work with Patrizio Angelini,](https://reader031.vdocuments.site/reader031/viewer/2022022807/5cde865488c993680f8d29bb/html5/thumbnails/27.jpg)
Windrose Planarity
Two directions:
q-constrained graph (G ,Q):• G : undirected planar graph• Q: partition of all neighbors of v into
↗
v ,↖
v ,↙
v , and↘
v .
↗
v↖
v
↘
v↙
v
A q-constrained graph is windrose planar :• no crossings• all edges are xy -monotone curves• u ∈ ◦
v ⇒ u lies in the ◦-quadrant of v
v
![Page 28: Windrose Planarity - pub.informatik.uni-wuerzburg.de fileWindrose Planarity Embedding Graphs with Direction-Constrained Edges Published at SODA'16. Joint work with Patrizio Angelini,](https://reader031.vdocuments.site/reader031/viewer/2022022807/5cde865488c993680f8d29bb/html5/thumbnails/28.jpg)
Relationship to Upwards Planarity
Two directions:
q-constrained graph (G ,Q):• G : undirected planar graph• Q: partition of all neighbors of v into
↗
v ,↖
v ,↙
v , and↘
v .
↗
v↖
v
↘
v↙
v
A q-constrained graph is windrose planar :• no crossings• all edges are xy -monotone curves• u ∈ ◦
v ⇒ u lies in the ◦-quadrant of v
v
![Page 29: Windrose Planarity - pub.informatik.uni-wuerzburg.de fileWindrose Planarity Embedding Graphs with Direction-Constrained Edges Published at SODA'16. Joint work with Patrizio Angelini,](https://reader031.vdocuments.site/reader031/viewer/2022022807/5cde865488c993680f8d29bb/html5/thumbnails/29.jpg)
Relationship to Upwards Planarity
Two directions:
q-constrained graph (G ,Q):• G : undirected planar graph• Q: partition of all neighbors of v into
↗
v and↙
v
v
↗
v
↙
v
A q-constrained graph is windrose planar :• no crossings• all edges are xy -monotone curves• u ∈ ◦
v ⇒ u lies in the ◦-quadrant of v
![Page 30: Windrose Planarity - pub.informatik.uni-wuerzburg.de fileWindrose Planarity Embedding Graphs with Direction-Constrained Edges Published at SODA'16. Joint work with Patrizio Angelini,](https://reader031.vdocuments.site/reader031/viewer/2022022807/5cde865488c993680f8d29bb/html5/thumbnails/30.jpg)
Relationship to Upwards Planarity
One direction:
q-constrained graph (G ,Q):• G : undirected planar graph• Q: partition of all neighbors of v into
↗
v and↙
v
v
↗
v
↙
v
A q-constrained graph is windrose planar :• no crossings• all edges are xy -monotone curves• u ∈ ◦
v ⇒ u lies in the ◦-quadrant of v
![Page 31: Windrose Planarity - pub.informatik.uni-wuerzburg.de fileWindrose Planarity Embedding Graphs with Direction-Constrained Edges Published at SODA'16. Joint work with Patrizio Angelini,](https://reader031.vdocuments.site/reader031/viewer/2022022807/5cde865488c993680f8d29bb/html5/thumbnails/31.jpg)
Relationship to Upwards Planarity
q-constrained graph (G ,Q):• G : undirected planar graph• Q: partition of all neighbors of v into
↑v and
↓v
A q-constrained graph is windrose planar :• no crossings• all edges are xy -monotone curves• u ∈ ◦
v ⇒ u lies in the ◦-quadrant of v
v
↑v
↓v
One direction:
![Page 32: Windrose Planarity - pub.informatik.uni-wuerzburg.de fileWindrose Planarity Embedding Graphs with Direction-Constrained Edges Published at SODA'16. Joint work with Patrizio Angelini,](https://reader031.vdocuments.site/reader031/viewer/2022022807/5cde865488c993680f8d29bb/html5/thumbnails/32.jpg)
Relationship to Upwards Planarity
directed graph
One direction:
A directed graph is upwards planar :
• all edges are y -monotone curves directed upwards
• no crossings
v
↑v
↓v
![Page 33: Windrose Planarity - pub.informatik.uni-wuerzburg.de fileWindrose Planarity Embedding Graphs with Direction-Constrained Edges Published at SODA'16. Joint work with Patrizio Angelini,](https://reader031.vdocuments.site/reader031/viewer/2022022807/5cde865488c993680f8d29bb/html5/thumbnails/33.jpg)
Relationship to Upwards Planarity
directed graph
One direction:
A directed graph is upwards planar :
• all edges are y -monotone curves directed upwards
• no crossings
v
↑v
↓v
Testing Windrose Planarityis NP-complete
Theorem.
![Page 34: Windrose Planarity - pub.informatik.uni-wuerzburg.de fileWindrose Planarity Embedding Graphs with Direction-Constrained Edges Published at SODA'16. Joint work with Patrizio Angelini,](https://reader031.vdocuments.site/reader031/viewer/2022022807/5cde865488c993680f8d29bb/html5/thumbnails/34.jpg)
Angular Drawing
Angle categories: 0◦, 90◦, 180◦, 270◦, and 360◦
![Page 35: Windrose Planarity - pub.informatik.uni-wuerzburg.de fileWindrose Planarity Embedding Graphs with Direction-Constrained Edges Published at SODA'16. Joint work with Patrizio Angelini,](https://reader031.vdocuments.site/reader031/viewer/2022022807/5cde865488c993680f8d29bb/html5/thumbnails/35.jpg)
Angular Drawing
Angle categories: 0◦, 90◦, 180◦, 270◦, and 360◦
![Page 36: Windrose Planarity - pub.informatik.uni-wuerzburg.de fileWindrose Planarity Embedding Graphs with Direction-Constrained Edges Published at SODA'16. Joint work with Patrizio Angelini,](https://reader031.vdocuments.site/reader031/viewer/2022022807/5cde865488c993680f8d29bb/html5/thumbnails/36.jpg)
Angular Drawing
Angle categories: 0◦, 90◦, 180◦, 270◦, and 360◦
![Page 37: Windrose Planarity - pub.informatik.uni-wuerzburg.de fileWindrose Planarity Embedding Graphs with Direction-Constrained Edges Published at SODA'16. Joint work with Patrizio Angelini,](https://reader031.vdocuments.site/reader031/viewer/2022022807/5cde865488c993680f8d29bb/html5/thumbnails/37.jpg)
Angular Drawing
Angle categories: 0◦, 90◦, 180◦, 270◦, and 360◦
![Page 38: Windrose Planarity - pub.informatik.uni-wuerzburg.de fileWindrose Planarity Embedding Graphs with Direction-Constrained Edges Published at SODA'16. Joint work with Patrizio Angelini,](https://reader031.vdocuments.site/reader031/viewer/2022022807/5cde865488c993680f8d29bb/html5/thumbnails/38.jpg)
Angular Drawing
Angle categories: 0◦, 90◦, 180◦, 270◦, and 360◦
0◦
![Page 39: Windrose Planarity - pub.informatik.uni-wuerzburg.de fileWindrose Planarity Embedding Graphs with Direction-Constrained Edges Published at SODA'16. Joint work with Patrizio Angelini,](https://reader031.vdocuments.site/reader031/viewer/2022022807/5cde865488c993680f8d29bb/html5/thumbnails/39.jpg)
Angular Drawing
Angle categories: 0◦, 90◦, 180◦, 270◦, and 360◦
0◦
![Page 40: Windrose Planarity - pub.informatik.uni-wuerzburg.de fileWindrose Planarity Embedding Graphs with Direction-Constrained Edges Published at SODA'16. Joint work with Patrizio Angelini,](https://reader031.vdocuments.site/reader031/viewer/2022022807/5cde865488c993680f8d29bb/html5/thumbnails/40.jpg)
Angular Drawing
Angle categories: 0◦, 90◦, 180◦, 270◦, and 360◦
90◦
0◦90◦
![Page 41: Windrose Planarity - pub.informatik.uni-wuerzburg.de fileWindrose Planarity Embedding Graphs with Direction-Constrained Edges Published at SODA'16. Joint work with Patrizio Angelini,](https://reader031.vdocuments.site/reader031/viewer/2022022807/5cde865488c993680f8d29bb/html5/thumbnails/41.jpg)
Angular Drawing
Angle categories: 0◦, 90◦, 180◦, 270◦, and 360◦
90◦
0◦90◦
![Page 42: Windrose Planarity - pub.informatik.uni-wuerzburg.de fileWindrose Planarity Embedding Graphs with Direction-Constrained Edges Published at SODA'16. Joint work with Patrizio Angelini,](https://reader031.vdocuments.site/reader031/viewer/2022022807/5cde865488c993680f8d29bb/html5/thumbnails/42.jpg)
Angular Drawing
Angle categories: 0◦, 90◦, 180◦, 270◦, and 360◦
90◦
0◦90◦
180◦
![Page 43: Windrose Planarity - pub.informatik.uni-wuerzburg.de fileWindrose Planarity Embedding Graphs with Direction-Constrained Edges Published at SODA'16. Joint work with Patrizio Angelini,](https://reader031.vdocuments.site/reader031/viewer/2022022807/5cde865488c993680f8d29bb/html5/thumbnails/43.jpg)
Angular Drawing
Angle categories: 0◦, 90◦, 180◦, 270◦, and 360◦
90◦
0◦90◦
180◦
![Page 44: Windrose Planarity - pub.informatik.uni-wuerzburg.de fileWindrose Planarity Embedding Graphs with Direction-Constrained Edges Published at SODA'16. Joint work with Patrizio Angelini,](https://reader031.vdocuments.site/reader031/viewer/2022022807/5cde865488c993680f8d29bb/html5/thumbnails/44.jpg)
Angular Drawing
Angle categories: 0◦, 90◦, 180◦, 270◦, and 360◦
90◦
270◦
0◦90◦
180◦
![Page 45: Windrose Planarity - pub.informatik.uni-wuerzburg.de fileWindrose Planarity Embedding Graphs with Direction-Constrained Edges Published at SODA'16. Joint work with Patrizio Angelini,](https://reader031.vdocuments.site/reader031/viewer/2022022807/5cde865488c993680f8d29bb/html5/thumbnails/45.jpg)
Angular Drawing
Angle categories: 0◦, 90◦, 180◦, 270◦, and 360◦
90◦
270◦
0◦90◦
180◦
![Page 46: Windrose Planarity - pub.informatik.uni-wuerzburg.de fileWindrose Planarity Embedding Graphs with Direction-Constrained Edges Published at SODA'16. Joint work with Patrizio Angelini,](https://reader031.vdocuments.site/reader031/viewer/2022022807/5cde865488c993680f8d29bb/html5/thumbnails/46.jpg)
Angular Drawing
Angle categories: 0◦, 90◦, 180◦, 270◦, and 360◦
90◦
270◦
360◦
0◦90◦
180◦
![Page 47: Windrose Planarity - pub.informatik.uni-wuerzburg.de fileWindrose Planarity Embedding Graphs with Direction-Constrained Edges Published at SODA'16. Joint work with Patrizio Angelini,](https://reader031.vdocuments.site/reader031/viewer/2022022807/5cde865488c993680f8d29bb/html5/thumbnails/47.jpg)
Angular Drawing
Angle categories: 0◦, 90◦, 180◦, 270◦, and 360◦
90◦
90◦ 180◦
270◦
0◦
180◦
0◦
360◦
0◦
270◦
90◦
0◦270◦
0◦90◦
180◦
90◦180◦
90◦
90◦
![Page 48: Windrose Planarity - pub.informatik.uni-wuerzburg.de fileWindrose Planarity Embedding Graphs with Direction-Constrained Edges Published at SODA'16. Joint work with Patrizio Angelini,](https://reader031.vdocuments.site/reader031/viewer/2022022807/5cde865488c993680f8d29bb/html5/thumbnails/48.jpg)
Angular Drawing
Angle categories: 0◦, 90◦, 180◦, 270◦, and 360◦
90◦
90◦ 180◦
270◦
0◦
180◦
0◦
360◦
0◦
270◦
90◦
0◦270◦
0◦90◦
180◦
90◦180◦
90◦
90◦
Labeled graph (G ,A): G plane graph, A labeling of angles
![Page 49: Windrose Planarity - pub.informatik.uni-wuerzburg.de fileWindrose Planarity Embedding Graphs with Direction-Constrained Edges Published at SODA'16. Joint work with Patrizio Angelini,](https://reader031.vdocuments.site/reader031/viewer/2022022807/5cde865488c993680f8d29bb/html5/thumbnails/49.jpg)
Angular Drawing
Angle categories: 0◦, 90◦, 180◦, 270◦, and 360◦
90◦
90◦ 180◦
270◦
0◦
180◦
0◦
360◦
0◦
270◦
90◦
0◦270◦
0◦90◦
180◦
90◦180◦
90◦
90◦
Labeled graph (G ,A): G plane graph, A labeling of anglesAngular drawing : end of segments have slopes ≈ ±1
![Page 50: Windrose Planarity - pub.informatik.uni-wuerzburg.de fileWindrose Planarity Embedding Graphs with Direction-Constrained Edges Published at SODA'16. Joint work with Patrizio Angelini,](https://reader031.vdocuments.site/reader031/viewer/2022022807/5cde865488c993680f8d29bb/html5/thumbnails/50.jpg)
Angular Drawing
Angle categories: 0◦, 90◦, 180◦, 270◦, and 360◦
Labeled graph (G ,A): G plane graph, A labeling of anglesAngular drawing : end of segments have slopes ≈ ±1
90◦
90◦ 180◦
270◦
0◦
180◦0◦
270◦
90◦
0◦
270◦
0◦90◦
180◦
90◦
180◦90◦
90◦
0◦
360◦
![Page 51: Windrose Planarity - pub.informatik.uni-wuerzburg.de fileWindrose Planarity Embedding Graphs with Direction-Constrained Edges Published at SODA'16. Joint work with Patrizio Angelini,](https://reader031.vdocuments.site/reader031/viewer/2022022807/5cde865488c993680f8d29bb/html5/thumbnails/51.jpg)
Angular Drawing
Angle categories: 0◦, 90◦, 180◦, 270◦, and 360◦
Labeled graph (G ,A): G plane graph, A labeling of anglesAngular drawing : end of segments have slopes ≈ ±1
90◦
90◦180◦
270◦
0◦
180◦0◦
270◦
90◦
0◦
270◦
0◦90◦
180◦
90◦
180◦90◦
90◦
0◦
360◦
![Page 52: Windrose Planarity - pub.informatik.uni-wuerzburg.de fileWindrose Planarity Embedding Graphs with Direction-Constrained Edges Published at SODA'16. Joint work with Patrizio Angelini,](https://reader031.vdocuments.site/reader031/viewer/2022022807/5cde865488c993680f8d29bb/html5/thumbnails/52.jpg)
Angular Drawing
(G ,A) admits angular drawing if:
Angle categories: 0◦, 90◦, 180◦, 270◦, and 360◦
Labeled graph (G ,A): G plane graph, A labeling of anglesAngular drawing : end of segments have slopes ≈ ±1
90◦
90◦180◦
270◦
0◦
180◦0◦
270◦
90◦
0◦
270◦
0◦90◦
180◦
90◦
180◦90◦
90◦
0◦
360◦
![Page 53: Windrose Planarity - pub.informatik.uni-wuerzburg.de fileWindrose Planarity Embedding Graphs with Direction-Constrained Edges Published at SODA'16. Joint work with Patrizio Angelini,](https://reader031.vdocuments.site/reader031/viewer/2022022807/5cde865488c993680f8d29bb/html5/thumbnails/53.jpg)
Angular Drawing
(G ,A) admits angular drawing if:• Vertex condition: sum of angle cat. at vertex is 360◦
Angle categories: 0◦, 90◦, 180◦, 270◦, and 360◦
Labeled graph (G ,A): G plane graph, A labeling of anglesAngular drawing : end of segments have slopes ≈ ±1
90◦
90◦180◦
270◦
0◦
180◦0◦
270◦
90◦
0◦
270◦
0◦90◦
180◦
90◦
180◦90◦
90◦
0◦
360◦
![Page 54: Windrose Planarity - pub.informatik.uni-wuerzburg.de fileWindrose Planarity Embedding Graphs with Direction-Constrained Edges Published at SODA'16. Joint work with Patrizio Angelini,](https://reader031.vdocuments.site/reader031/viewer/2022022807/5cde865488c993680f8d29bb/html5/thumbnails/54.jpg)
Angular Drawing
(G ,A) admits angular drawing if:• Vertex condition: sum of angle cat. at vertex is 360◦
Angle categories: 0◦, 90◦, 180◦, 270◦, and 360◦
Labeled graph (G ,A): G plane graph, A labeling of anglesAngular drawing : end of segments have slopes ≈ ±1
90◦
90◦180◦
270◦
0◦
180◦0◦
270◦
90◦
0◦
270◦
0◦90◦
180◦
90◦
180◦90◦
90◦
0◦
360◦
![Page 55: Windrose Planarity - pub.informatik.uni-wuerzburg.de fileWindrose Planarity Embedding Graphs with Direction-Constrained Edges Published at SODA'16. Joint work with Patrizio Angelini,](https://reader031.vdocuments.site/reader031/viewer/2022022807/5cde865488c993680f8d29bb/html5/thumbnails/55.jpg)
Angular Drawing
(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 oflength k is k · 180◦ − 360◦
Angle categories: 0◦, 90◦, 180◦, 270◦, and 360◦
Labeled graph (G ,A): G plane graph, A labeling of anglesAngular drawing : end of segments have slopes ≈ ±1
90◦
90◦180◦
270◦
0◦
180◦0◦
270◦
90◦
0◦
270◦
0◦90◦
180◦
90◦
180◦90◦
90◦
0◦
360◦
![Page 56: Windrose Planarity - pub.informatik.uni-wuerzburg.de fileWindrose Planarity Embedding Graphs with Direction-Constrained Edges Published at SODA'16. Joint work with Patrizio Angelini,](https://reader031.vdocuments.site/reader031/viewer/2022022807/5cde865488c993680f8d29bb/html5/thumbnails/56.jpg)
Angular Drawing
(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 oflength k is k · 180◦ − 360◦
Angle categories: 0◦, 90◦, 180◦, 270◦, and 360◦
Labeled graph (G ,A): G plane graph, A labeling of anglesAngular drawing : end of segments have slopes ≈ ±1
90◦
90◦180◦
270◦
0◦
180◦0◦
270◦
90◦
0◦
270◦
0◦90◦
180◦
90◦
180◦90◦
90◦
0◦
360◦
![Page 57: Windrose Planarity - pub.informatik.uni-wuerzburg.de fileWindrose Planarity Embedding Graphs with Direction-Constrained Edges Published at SODA'16. Joint work with Patrizio Angelini,](https://reader031.vdocuments.site/reader031/viewer/2022022807/5cde865488c993680f8d29bb/html5/thumbnails/57.jpg)
Angular Drawing
(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 oflength k is k · 180◦ − 360◦
Angle categories: 0◦, 90◦, 180◦, 270◦, and 360◦
Labeled graph (G ,A): G plane graph, A labeling of anglesAngular drawing : end of segments have slopes ≈ ±1
90◦
90◦180◦
270◦
0◦
180◦0◦
270◦
90◦
0◦
270◦
0◦90◦
180◦
90◦
180◦90◦
90◦
0◦
360◦
⇒ A is angular labeling
![Page 58: Windrose Planarity - pub.informatik.uni-wuerzburg.de fileWindrose Planarity Embedding Graphs with Direction-Constrained Edges Published at SODA'16. Joint work with Patrizio Angelini,](https://reader031.vdocuments.site/reader031/viewer/2022022807/5cde865488c993680f8d29bb/html5/thumbnails/58.jpg)
Angular Drawing
(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 oflength k is k · 180◦ − 360◦
Angle categories: 0◦, 90◦, 180◦, 270◦, and 360◦
Labeled graph (G ,A): G plane graph, A labeling of anglesAngular drawing : end of segments have slopes ≈ ±1
90◦
90◦180◦
270◦
0◦
180◦0◦
270◦
90◦
0◦
270◦
0◦90◦
180◦
90◦
180◦90◦
90◦
0◦
360◦
⇒ A is angular labeling
angular labeling A⇒
uniqueq-constraints QA
![Page 59: Windrose Planarity - pub.informatik.uni-wuerzburg.de fileWindrose Planarity Embedding Graphs with Direction-Constrained Edges Published at SODA'16. Joint work with Patrizio Angelini,](https://reader031.vdocuments.site/reader031/viewer/2022022807/5cde865488c993680f8d29bb/html5/thumbnails/59.jpg)
Angular Drawing
(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 oflength k is k · 180◦ − 360◦
Angle categories: 0◦, 90◦, 180◦, 270◦, and 360◦
Labeled graph (G ,A): G plane graph, A labeling of anglesAngular drawing : end of segments have slopes ≈ ±1
90◦
90◦180◦
270◦
0◦
180◦0◦
270◦
90◦
0◦
270◦
0◦90◦
180◦
90◦
180◦90◦
90◦
0◦
360◦
⇒ A is angular labeling
angular labeling A⇒
uniqueq-constraints QA
![Page 60: Windrose Planarity - pub.informatik.uni-wuerzburg.de fileWindrose Planarity Embedding Graphs with Direction-Constrained Edges Published at SODA'16. Joint work with Patrizio Angelini,](https://reader031.vdocuments.site/reader031/viewer/2022022807/5cde865488c993680f8d29bb/html5/thumbnails/60.jpg)
Angular Drawing
(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 oflength k is k · 180◦ − 360◦
Angle categories: 0◦, 90◦, 180◦, 270◦, and 360◦
Labeled graph (G ,A): G plane graph, A labeling of anglesAngular drawing : end of segments have slopes ≈ ±1
90◦
90◦180◦
270◦
0◦
180◦0◦
270◦
90◦
0◦
270◦
0◦90◦
180◦
90◦
180◦90◦
90◦
0◦
360◦
⇒ A is angular labeling
angular labeling A⇒
uniqueq-constraints QA
q-constraints Q+ large-angle assignment L⇒
uniqueangular labeling AQ,L
![Page 61: Windrose Planarity - pub.informatik.uni-wuerzburg.de fileWindrose Planarity Embedding Graphs with Direction-Constrained Edges Published at SODA'16. Joint work with Patrizio Angelini,](https://reader031.vdocuments.site/reader031/viewer/2022022807/5cde865488c993680f8d29bb/html5/thumbnails/61.jpg)
Angular Drawing
(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 oflength k is k · 180◦ − 360◦
Angle categories: 0◦, 90◦, 180◦, 270◦, and 360◦
Labeled graph (G ,A): G plane graph, A labeling of anglesAngular drawing : end of segments have slopes ≈ ±1
90◦
90◦180◦
270◦
0◦
180◦0◦
270◦
90◦
0◦
270◦
0◦90◦
180◦
90◦
180◦90◦
90◦
0◦
360◦
⇒ A is angular labeling
angular labeling A⇒
uniqueq-constraints QA
q-constraints Q+ large-angle assignment L⇒
uniqueangular labeling AQ,L
angular drawing= windrose planar drawing
![Page 62: Windrose Planarity - pub.informatik.uni-wuerzburg.de fileWindrose Planarity Embedding Graphs with Direction-Constrained Edges Published at SODA'16. Joint work with Patrizio Angelini,](https://reader031.vdocuments.site/reader031/viewer/2022022807/5cde865488c993680f8d29bb/html5/thumbnails/62.jpg)
Triangulated Graphs0 180 0
![Page 63: Windrose Planarity - pub.informatik.uni-wuerzburg.de fileWindrose Planarity Embedding Graphs with Direction-Constrained Edges Published at SODA'16. Joint work with Patrizio Angelini,](https://reader031.vdocuments.site/reader031/viewer/2022022807/5cde865488c993680f8d29bb/html5/thumbnails/63.jpg)
Triangulated Graphs0 180 0
![Page 64: Windrose Planarity - pub.informatik.uni-wuerzburg.de fileWindrose Planarity Embedding Graphs with Direction-Constrained Edges Published at SODA'16. Joint work with Patrizio Angelini,](https://reader031.vdocuments.site/reader031/viewer/2022022807/5cde865488c993680f8d29bb/html5/thumbnails/64.jpg)
Triangulated Graphs0 90 90
![Page 65: Windrose Planarity - pub.informatik.uni-wuerzburg.de fileWindrose Planarity Embedding Graphs with Direction-Constrained Edges Published at SODA'16. Joint work with Patrizio Angelini,](https://reader031.vdocuments.site/reader031/viewer/2022022807/5cde865488c993680f8d29bb/html5/thumbnails/65.jpg)
Triangulated Graphs
• No (int.) > 180◦ angle categories0 90 90
![Page 66: Windrose Planarity - pub.informatik.uni-wuerzburg.de fileWindrose Planarity Embedding Graphs with Direction-Constrained Edges Published at SODA'16. Joint work with Patrizio Angelini,](https://reader031.vdocuments.site/reader031/viewer/2022022807/5cde865488c993680f8d29bb/html5/thumbnails/66.jpg)
Triangulated Graphs
• No (int.) > 180◦ angle categories• At least one 0◦ angle category per (int.) face
0 90 90
![Page 67: Windrose Planarity - pub.informatik.uni-wuerzburg.de fileWindrose Planarity Embedding Graphs with Direction-Constrained Edges Published at SODA'16. Joint work with Patrizio Angelini,](https://reader031.vdocuments.site/reader031/viewer/2022022807/5cde865488c993680f8d29bb/html5/thumbnails/67.jpg)
Triangulated Graphs
• No (int.) > 180◦ angle categories• At least one 0◦ angle category per (int.) face
0 90 90
![Page 68: Windrose Planarity - pub.informatik.uni-wuerzburg.de fileWindrose Planarity Embedding Graphs with Direction-Constrained Edges Published at SODA'16. Joint work with Patrizio Angelini,](https://reader031.vdocuments.site/reader031/viewer/2022022807/5cde865488c993680f8d29bb/html5/thumbnails/68.jpg)
Triangulated Graphs
• No (int.) > 180◦ angle categories• At least one 0◦ angle category per (int.) face
0 90 90
G↑
![Page 69: Windrose Planarity - pub.informatik.uni-wuerzburg.de fileWindrose Planarity Embedding Graphs with Direction-Constrained Edges Published at SODA'16. Joint work with Patrizio Angelini,](https://reader031.vdocuments.site/reader031/viewer/2022022807/5cde865488c993680f8d29bb/html5/thumbnails/69.jpg)
Triangulated Graphs
• No (int.) > 180◦ angle categories• At least one 0◦ angle category per (int.) face
0 90 90
G↑
G↑ is acyclic.
Lemma.Let (G ,AQ) be a triangulated angular labeled graph. Then,
![Page 70: Windrose Planarity - pub.informatik.uni-wuerzburg.de fileWindrose Planarity Embedding Graphs with Direction-Constrained Edges Published at SODA'16. Joint work with Patrizio Angelini,](https://reader031.vdocuments.site/reader031/viewer/2022022807/5cde865488c993680f8d29bb/html5/thumbnails/70.jpg)
Triangulated Graphs
• No (int.) > 180◦ angle categories• At least one 0◦ angle category per (int.) face
0 90 90
G↑
G↑ is acyclic and has no internal sources or sinks.
Lemma.Let (G ,AQ) be a triangulated angular labeled graph. Then,
270◦270◦
![Page 71: Windrose Planarity - pub.informatik.uni-wuerzburg.de fileWindrose Planarity Embedding Graphs with Direction-Constrained Edges Published at SODA'16. Joint work with Patrizio Angelini,](https://reader031.vdocuments.site/reader031/viewer/2022022807/5cde865488c993680f8d29bb/html5/thumbnails/71.jpg)
Triangulated Graphs
G→
• No (int.) > 180◦ angle categories• At least one 0◦ angle category per (int.) face
0 90 90
G↑ is acyclic and has no internal sources or sinks.
Lemma.Let (G ,AQ) be a triangulated angular labeled graph. Then,
270◦270◦
![Page 72: Windrose Planarity - pub.informatik.uni-wuerzburg.de fileWindrose Planarity Embedding Graphs with Direction-Constrained Edges Published at SODA'16. Joint work with Patrizio Angelini,](https://reader031.vdocuments.site/reader031/viewer/2022022807/5cde865488c993680f8d29bb/html5/thumbnails/72.jpg)
Triangulated Graphs
G→
• No (int.) > 180◦ angle categories• At least one 0◦ angle category per (int.) face
0 90 90
Lemma.
G↑ and G→ are acyclic and have no internal sources or sinks.
Let (G ,AQ) be a triangulated angular labeled graph. Then,
![Page 73: Windrose Planarity - pub.informatik.uni-wuerzburg.de fileWindrose Planarity Embedding Graphs with Direction-Constrained Edges Published at SODA'16. Joint work with Patrizio Angelini,](https://reader031.vdocuments.site/reader031/viewer/2022022807/5cde865488c993680f8d29bb/html5/thumbnails/73.jpg)
Triangulated Graphs
• No (int.) > 180◦ angle categories• At least one 0◦ angle category per (int.) face
0 90 90
Lemma.
G↑ and G→ are acyclic and have no internal sources or sinks.
Let (G ,AQ) be a triangulated angular labeled graph. Then,
G
![Page 74: Windrose Planarity - pub.informatik.uni-wuerzburg.de fileWindrose Planarity Embedding Graphs with Direction-Constrained Edges Published at SODA'16. Joint work with Patrizio Angelini,](https://reader031.vdocuments.site/reader031/viewer/2022022807/5cde865488c993680f8d29bb/html5/thumbnails/74.jpg)
Triangulated Graphs
• No (int.) > 180◦ angle categories• At least one 0◦ angle category per (int.) face
0 90 90
Lemma.
G↑ and G→ are acyclic and have no internal sources or sinks.
Let (G ,AQ) be a triangulated angular labeled graph. Then,
G
internally
![Page 75: Windrose Planarity - pub.informatik.uni-wuerzburg.de fileWindrose Planarity Embedding Graphs with Direction-Constrained Edges Published at SODA'16. Joint work with Patrizio Angelini,](https://reader031.vdocuments.site/reader031/viewer/2022022807/5cde865488c993680f8d29bb/html5/thumbnails/75.jpg)
Triangulated Graphs
• No (int.) > 180◦ angle categories• At least one 0◦ angle category per (int.) face
0 90 90
Lemma.
G↑ and G→ are acyclic and have no internal sources or sinks.
Let (G ,AQ) be a triangulated angular labeled graph. Then,
G
internally
What if there are no (int.)180◦ angle categories?
![Page 76: Windrose Planarity - pub.informatik.uni-wuerzburg.de fileWindrose Planarity Embedding Graphs with Direction-Constrained Edges Published at SODA'16. Joint work with Patrizio Angelini,](https://reader031.vdocuments.site/reader031/viewer/2022022807/5cde865488c993680f8d29bb/html5/thumbnails/76.jpg)
Quasi-triangulated Graphs
What if there are no (int.) 180◦ angle categories?
![Page 77: Windrose Planarity - pub.informatik.uni-wuerzburg.de fileWindrose Planarity Embedding Graphs with Direction-Constrained Edges Published at SODA'16. Joint work with Patrizio Angelini,](https://reader031.vdocuments.site/reader031/viewer/2022022807/5cde865488c993680f8d29bb/html5/thumbnails/77.jpg)
Quasi-triangulated Graphs
≤ 90◦
4 vertices outside
wN
wWwS
What if there are no (int.) 180◦ angle categories?
![Page 78: Windrose Planarity - pub.informatik.uni-wuerzburg.de fileWindrose Planarity Embedding Graphs with Direction-Constrained Edges Published at SODA'16. Joint work with Patrizio Angelini,](https://reader031.vdocuments.site/reader031/viewer/2022022807/5cde865488c993680f8d29bb/html5/thumbnails/78.jpg)
Quasi-triangulated Graphs
≤ 90◦
4 vertices outside
wN
wWwS
What if there are no (int.) 180◦ angle categories?
wE
![Page 79: Windrose Planarity - pub.informatik.uni-wuerzburg.de fileWindrose Planarity Embedding Graphs with Direction-Constrained Edges Published at SODA'16. Joint work with Patrizio Angelini,](https://reader031.vdocuments.site/reader031/viewer/2022022807/5cde865488c993680f8d29bb/html5/thumbnails/79.jpg)
Quasi-triangulated Graphs
What if there are no (int.) 180◦ angle categories?
wN
wWwS
wE
G→
![Page 80: Windrose Planarity - pub.informatik.uni-wuerzburg.de fileWindrose Planarity Embedding Graphs with Direction-Constrained Edges Published at SODA'16. Joint work with Patrizio Angelini,](https://reader031.vdocuments.site/reader031/viewer/2022022807/5cde865488c993680f8d29bb/html5/thumbnails/80.jpg)
Quasi-triangulated Graphs
What if there are no (int.) 180◦ angle categories?
wN
wWwS
wE
• topological order on G→:
G→
x-coordinates
![Page 81: Windrose Planarity - pub.informatik.uni-wuerzburg.de fileWindrose Planarity Embedding Graphs with Direction-Constrained Edges Published at SODA'16. Joint work with Patrizio Angelini,](https://reader031.vdocuments.site/reader031/viewer/2022022807/5cde865488c993680f8d29bb/html5/thumbnails/81.jpg)
Quasi-triangulated Graphs
What if there are no (int.) 180◦ angle categories?
wN
wWwS
wE
• topological order on G→:
G→
1
2
3
4
5
67
8
9
10
11
12
13
14
x-coordinates
![Page 82: Windrose Planarity - pub.informatik.uni-wuerzburg.de fileWindrose Planarity Embedding Graphs with Direction-Constrained Edges Published at SODA'16. Joint work with Patrizio Angelini,](https://reader031.vdocuments.site/reader031/viewer/2022022807/5cde865488c993680f8d29bb/html5/thumbnails/82.jpg)
Quasi-triangulated Graphs
1 3 5 7 9 11 13
What if there are no (int.) 180◦ angle categories?
wN
wWwS
wE
• topological order on G→:
G→
1
2
3
4
5
67
8
9
10
11
12
13
14
x-coordinates
![Page 83: Windrose Planarity - pub.informatik.uni-wuerzburg.de fileWindrose Planarity Embedding Graphs with Direction-Constrained Edges Published at SODA'16. Joint work with Patrizio Angelini,](https://reader031.vdocuments.site/reader031/viewer/2022022807/5cde865488c993680f8d29bb/html5/thumbnails/83.jpg)
Quasi-triangulated Graphs
1 3 5 7 9 11 13
What if there are no (int.) 180◦ angle categories?• topological order on G→:
wN
wWwS
wE
G↑
x-coordinates
![Page 84: Windrose Planarity - pub.informatik.uni-wuerzburg.de fileWindrose Planarity Embedding Graphs with Direction-Constrained Edges Published at SODA'16. Joint work with Patrizio Angelini,](https://reader031.vdocuments.site/reader031/viewer/2022022807/5cde865488c993680f8d29bb/html5/thumbnails/84.jpg)
Quasi-triangulated Graphs
1 3 5 7 9 11 13
What if there are no (int.) 180◦ angle categories?• topological order on G→:
wN
wWwS
wE
G↑
• topological order on G↑:
x-coordinatesy -coordinates
![Page 85: Windrose Planarity - pub.informatik.uni-wuerzburg.de fileWindrose Planarity Embedding Graphs with Direction-Constrained Edges Published at SODA'16. Joint work with Patrizio Angelini,](https://reader031.vdocuments.site/reader031/viewer/2022022807/5cde865488c993680f8d29bb/html5/thumbnails/85.jpg)
Quasi-triangulated Graphs
1 3 5 7 9 11 13
What if there are no (int.) 180◦ angle categories?• topological order on G→:
wN
wWwS
wE
G↑
• topological order on G↑:
2
14
3
10
5
98
12
4
7
11
6
1
13
x-coordinatesy -coordinates
![Page 86: Windrose Planarity - pub.informatik.uni-wuerzburg.de fileWindrose Planarity Embedding Graphs with Direction-Constrained Edges Published at SODA'16. Joint work with Patrizio Angelini,](https://reader031.vdocuments.site/reader031/viewer/2022022807/5cde865488c993680f8d29bb/html5/thumbnails/86.jpg)
Quasi-triangulated Graphs
1 3 5 7 9 11 13
What if there are no (int.) 180◦ angle categories?• topological order on G→:
wN
wWwS
wE
G↑
• topological order on G↑:
2
14
3
10
5
98
12
4
7
11
6
1
13
x-coordinatesy -coordinates
![Page 87: Windrose Planarity - pub.informatik.uni-wuerzburg.de fileWindrose Planarity Embedding Graphs with Direction-Constrained Edges Published at SODA'16. Joint work with Patrizio Angelini,](https://reader031.vdocuments.site/reader031/viewer/2022022807/5cde865488c993680f8d29bb/html5/thumbnails/87.jpg)
Quasi-triangulated Graphs
1 3 5 7 9 11 13
What if there are no (int.) 180◦ angle categories?• topological order on G→:
wN
wWwS
wE
G↑
• topological order on G↑:
2
14
3
10
5
98
12
4
7
11
6
1
13
x-coordinatesy -coordinates
![Page 88: Windrose Planarity - pub.informatik.uni-wuerzburg.de fileWindrose Planarity Embedding Graphs with Direction-Constrained Edges Published at SODA'16. Joint work with Patrizio Angelini,](https://reader031.vdocuments.site/reader031/viewer/2022022807/5cde865488c993680f8d29bb/html5/thumbnails/88.jpg)
Quasi-triangulated Graphs
1 3 5 7 9 11 13
What if there are no (int.) 180◦ angle categories?• topological order on G→:
wN
wWwS
wE
G↑
• topological order on G↑:
2
14
3
10
5
98
12
4
7
11
6
1
13
x-coordinatesy -coordinates
![Page 89: Windrose Planarity - pub.informatik.uni-wuerzburg.de fileWindrose Planarity Embedding Graphs with Direction-Constrained Edges Published at SODA'16. Joint work with Patrizio Angelini,](https://reader031.vdocuments.site/reader031/viewer/2022022807/5cde865488c993680f8d29bb/html5/thumbnails/89.jpg)
Quasi-triangulated Graphs
1 3 5 7 9 11 13
What if there are no (int.) 180◦ angle categories?• topological order on G→:
wN
wWwS
wE
G↑
• topological order on G↑:
2
14
3
10
5
98
12
4
7
11
6
1
13
x-coordinatesy -coordinates
![Page 90: Windrose Planarity - pub.informatik.uni-wuerzburg.de fileWindrose Planarity Embedding Graphs with Direction-Constrained Edges Published at SODA'16. Joint work with Patrizio Angelini,](https://reader031.vdocuments.site/reader031/viewer/2022022807/5cde865488c993680f8d29bb/html5/thumbnails/90.jpg)
Quasi-triangulated Graphs
1 3 5 7 9 11 13
What if there are no (int.) 180◦ angle categories?• topological order on G→:• topological order on G↑:
x-coordinatesy -coordinates
![Page 91: Windrose Planarity - pub.informatik.uni-wuerzburg.de fileWindrose Planarity Embedding Graphs with Direction-Constrained Edges Published at SODA'16. Joint work with Patrizio Angelini,](https://reader031.vdocuments.site/reader031/viewer/2022022807/5cde865488c993680f8d29bb/html5/thumbnails/91.jpg)
Quasi-triangulated Graphs
1 3 5 7 9 11 13
What if there are no (int.) 180◦ angle categories?• topological order on G→:• topological order on G↑:
x-coordinatesy -coordinates
![Page 92: Windrose Planarity - pub.informatik.uni-wuerzburg.de fileWindrose Planarity Embedding Graphs with Direction-Constrained Edges Published at SODA'16. Joint work with Patrizio Angelini,](https://reader031.vdocuments.site/reader031/viewer/2022022807/5cde865488c993680f8d29bb/html5/thumbnails/92.jpg)
Quasi-triangulated Graphs
1 3 5 7 9 11 13
What if there are no (int.) 180◦ angle categories?• topological order on G→:• topological order on G↑:
0◦
x-coordinatesy -coordinates
![Page 93: Windrose Planarity - pub.informatik.uni-wuerzburg.de fileWindrose Planarity Embedding Graphs with Direction-Constrained Edges Published at SODA'16. Joint work with Patrizio Angelini,](https://reader031.vdocuments.site/reader031/viewer/2022022807/5cde865488c993680f8d29bb/html5/thumbnails/93.jpg)
Quasi-triangulated Graphs
1 3 5 7 9 11 13
What if there are no (int.) 180◦ angle categories?• topological order on G→:• topological order on G↑:
0◦
90◦
90◦
x-coordinatesy -coordinates
![Page 94: Windrose Planarity - pub.informatik.uni-wuerzburg.de fileWindrose Planarity Embedding Graphs with Direction-Constrained Edges Published at SODA'16. Joint work with Patrizio Angelini,](https://reader031.vdocuments.site/reader031/viewer/2022022807/5cde865488c993680f8d29bb/html5/thumbnails/94.jpg)
Quasi-triangulated Graphs
1 3 5 7 9 11 13
What if there are no (int.) 180◦ angle categories?• topological order on G→:• topological order on G↑:
0◦
90◦
90◦
x-coordinatesy -coordinates
![Page 95: Windrose Planarity - pub.informatik.uni-wuerzburg.de fileWindrose Planarity Embedding Graphs with Direction-Constrained Edges Published at SODA'16. Joint work with Patrizio Angelini,](https://reader031.vdocuments.site/reader031/viewer/2022022807/5cde865488c993680f8d29bb/html5/thumbnails/95.jpg)
Quasi-triangulated Graphs
1 3 5 7 9 11 13
What if there are no (int.) 180◦ angle categories?• topological order on G→:• topological order on G↑:
0◦
90◦
90◦
x-coordinatesy -coordinates
![Page 96: Windrose Planarity - pub.informatik.uni-wuerzburg.de fileWindrose Planarity Embedding Graphs with Direction-Constrained Edges Published at SODA'16. Joint work with Patrizio Angelini,](https://reader031.vdocuments.site/reader031/viewer/2022022807/5cde865488c993680f8d29bb/html5/thumbnails/96.jpg)
Quasi-triangulated Graphs
1 3 5 7 9 11 13
What if there are no (int.) 180◦ angle categories?• topological order on G→:• topological order on G↑:
0◦
90◦
90◦
Lemma.quasi-triangulated angular labeled graph (G ,AQ),all internal angles have category 0◦ or 90◦
⇒ straight-line windrose planar drawingon n × n grid in O(n) time
x-coordinatesy -coordinates
![Page 97: Windrose Planarity - pub.informatik.uni-wuerzburg.de fileWindrose Planarity Embedding Graphs with Direction-Constrained Edges Published at SODA'16. Joint work with Patrizio Angelini,](https://reader031.vdocuments.site/reader031/viewer/2022022807/5cde865488c993680f8d29bb/html5/thumbnails/97.jpg)
Triangulated graphs
Let (G ,AQ) be a triangulated angular labeled graph.
![Page 98: Windrose Planarity - pub.informatik.uni-wuerzburg.de fileWindrose Planarity Embedding Graphs with Direction-Constrained Edges Published at SODA'16. Joint work with Patrizio Angelini,](https://reader031.vdocuments.site/reader031/viewer/2022022807/5cde865488c993680f8d29bb/html5/thumbnails/98.jpg)
Triangulated graphs
Let (G ,AQ) be a triangulated angular labeled graph.
Task: Augment (G ,AQ) to a quasi-triangulated angular labeledgraph (G∗,AQ∗) without internal angle category 180◦.
![Page 99: Windrose Planarity - pub.informatik.uni-wuerzburg.de fileWindrose Planarity Embedding Graphs with Direction-Constrained Edges Published at SODA'16. Joint work with Patrizio Angelini,](https://reader031.vdocuments.site/reader031/viewer/2022022807/5cde865488c993680f8d29bb/html5/thumbnails/99.jpg)
Triangulated graphs
Let (G ,AQ) be a triangulated angular labeled graph.
Task: Augment (G ,AQ) to a quasi-triangulated angular labeledgraph (G∗,AQ∗) without internal angle category 180◦.
v
![Page 100: Windrose Planarity - pub.informatik.uni-wuerzburg.de fileWindrose Planarity Embedding Graphs with Direction-Constrained Edges Published at SODA'16. Joint work with Patrizio Angelini,](https://reader031.vdocuments.site/reader031/viewer/2022022807/5cde865488c993680f8d29bb/html5/thumbnails/100.jpg)
Triangulated graphs
Let (G ,AQ) be a triangulated angular labeled graph.
Task: Augment (G ,AQ) to a quasi-triangulated angular labeledgraph (G∗,AQ∗) without internal angle category 180◦.
v
u
![Page 101: Windrose Planarity - pub.informatik.uni-wuerzburg.de fileWindrose Planarity Embedding Graphs with Direction-Constrained Edges Published at SODA'16. Joint work with Patrizio Angelini,](https://reader031.vdocuments.site/reader031/viewer/2022022807/5cde865488c993680f8d29bb/html5/thumbnails/101.jpg)
Triangulated graphs
Let (G ,AQ) be a triangulated angular labeled graph.
Task: Augment (G ,AQ) to a quasi-triangulated angular labeledgraph (G∗,AQ∗) without internal angle category 180◦.
v
u
w
![Page 102: Windrose Planarity - pub.informatik.uni-wuerzburg.de fileWindrose Planarity Embedding Graphs with Direction-Constrained Edges Published at SODA'16. Joint work with Patrizio Angelini,](https://reader031.vdocuments.site/reader031/viewer/2022022807/5cde865488c993680f8d29bb/html5/thumbnails/102.jpg)
Triangulated graphs
Let (G ,AQ) be a triangulated angular labeled graph.
Task: Augment (G ,AQ) to a quasi-triangulated angular labeledgraph (G∗,AQ∗) without internal angle category 180◦.
v
u
w
![Page 103: Windrose Planarity - pub.informatik.uni-wuerzburg.de fileWindrose Planarity Embedding Graphs with Direction-Constrained Edges Published at SODA'16. Joint work with Patrizio Angelini,](https://reader031.vdocuments.site/reader031/viewer/2022022807/5cde865488c993680f8d29bb/html5/thumbnails/103.jpg)
Triangulated graphs
Let (G ,AQ) be a triangulated angular labeled graph.
v
u
w
• Assume that↗
u 6= ∅, ↗
w 6= ∅ (otherwise, set v = u/w)
![Page 104: Windrose Planarity - pub.informatik.uni-wuerzburg.de fileWindrose Planarity Embedding Graphs with Direction-Constrained Edges Published at SODA'16. Joint work with Patrizio Angelini,](https://reader031.vdocuments.site/reader031/viewer/2022022807/5cde865488c993680f8d29bb/html5/thumbnails/104.jpg)
Triangulated graphs
Let (G ,AQ) be a triangulated angular labeled graph.
v
u
w
• Assume that↗
u 6= ∅, ↗
w 6= ∅ (otherwise, set v = u/w)• if (u,w) not on outer face:
![Page 105: Windrose Planarity - pub.informatik.uni-wuerzburg.de fileWindrose Planarity Embedding Graphs with Direction-Constrained Edges Published at SODA'16. Joint work with Patrizio Angelini,](https://reader031.vdocuments.site/reader031/viewer/2022022807/5cde865488c993680f8d29bb/html5/thumbnails/105.jpg)
Triangulated graphs
Let (G ,AQ) be a triangulated angular labeled graph.
v
u
w
• Assume that↗
u 6= ∅, ↗
w 6= ∅ (otherwise, set v = u/w)• if (u,w) not on outer face:
![Page 106: Windrose Planarity - pub.informatik.uni-wuerzburg.de fileWindrose Planarity Embedding Graphs with Direction-Constrained Edges Published at SODA'16. Joint work with Patrizio Angelini,](https://reader031.vdocuments.site/reader031/viewer/2022022807/5cde865488c993680f8d29bb/html5/thumbnails/106.jpg)
Triangulated graphs
Let (G ,AQ) be a triangulated angular labeled graph.
v
u
w
• Assume that↗
u 6= ∅, ↗
w 6= ∅ (otherwise, set v = u/w)• if (u,w) not on outer face:
![Page 107: Windrose Planarity - pub.informatik.uni-wuerzburg.de fileWindrose Planarity Embedding Graphs with Direction-Constrained Edges Published at SODA'16. Joint work with Patrizio Angelini,](https://reader031.vdocuments.site/reader031/viewer/2022022807/5cde865488c993680f8d29bb/html5/thumbnails/107.jpg)
Triangulated graphs
Let (G ,AQ) be a triangulated angular labeled graph.
v
u
w
• Assume that↗
u 6= ∅, ↗
w 6= ∅ (otherwise, set v = u/w)• if (u,w) not on outer face:
![Page 108: Windrose Planarity - pub.informatik.uni-wuerzburg.de fileWindrose Planarity Embedding Graphs with Direction-Constrained Edges Published at SODA'16. Joint work with Patrizio Angelini,](https://reader031.vdocuments.site/reader031/viewer/2022022807/5cde865488c993680f8d29bb/html5/thumbnails/108.jpg)
Triangulated graphs
Let (G ,AQ) be a triangulated angular labeled graph.
v
u
w
• Assume that↗
u 6= ∅, ↗
w 6= ∅ (otherwise, set v = u/w)• if (u,w) not on outer face: • if (u,w) on outer face:
v
u
w
![Page 109: Windrose Planarity - pub.informatik.uni-wuerzburg.de fileWindrose Planarity Embedding Graphs with Direction-Constrained Edges Published at SODA'16. Joint work with Patrizio Angelini,](https://reader031.vdocuments.site/reader031/viewer/2022022807/5cde865488c993680f8d29bb/html5/thumbnails/109.jpg)
Triangulated graphs
Let (G ,AQ) be a triangulated angular labeled graph.
v
u
w
• Assume that↗
u 6= ∅, ↗
w 6= ∅ (otherwise, set v = u/w)• if (u,w) not on outer face: • if (u,w) on outer face:
v
u
w
![Page 110: Windrose Planarity - pub.informatik.uni-wuerzburg.de fileWindrose Planarity Embedding Graphs with Direction-Constrained Edges Published at SODA'16. Joint work with Patrizio Angelini,](https://reader031.vdocuments.site/reader031/viewer/2022022807/5cde865488c993680f8d29bb/html5/thumbnails/110.jpg)
Triangulated graphs
Let (G ,AQ) be a triangulated angular labeled graph.
v
u
w
• Assume that↗
u 6= ∅, ↗
w 6= ∅ (otherwise, set v = u/w)• if (u,w) not on outer face: • if (u,w) on outer face:
v
u
w• Add wN, wE, wS, and wW
![Page 111: Windrose Planarity - pub.informatik.uni-wuerzburg.de fileWindrose Planarity Embedding Graphs with Direction-Constrained Edges Published at SODA'16. Joint work with Patrizio Angelini,](https://reader031.vdocuments.site/reader031/viewer/2022022807/5cde865488c993680f8d29bb/html5/thumbnails/111.jpg)
Triangulated graphs
Let (G ,AQ) be a triangulated angular labeled graph.
v
u
w
• Assume that↗
u 6= ∅, ↗
w 6= ∅ (otherwise, set v = u/w)• if (u,w) not on outer face: • if (u,w) on outer face:
v
u
w• Add wN, wE, wS, and wW
![Page 112: Windrose Planarity - pub.informatik.uni-wuerzburg.de fileWindrose Planarity Embedding Graphs with Direction-Constrained Edges Published at SODA'16. Joint work with Patrizio Angelini,](https://reader031.vdocuments.site/reader031/viewer/2022022807/5cde865488c993680f8d29bb/html5/thumbnails/112.jpg)
Triangulated graphs
Let (G ,AQ) be a triangulated angular labeled graph.
v
u
w
• Assume that↗
u 6= ∅, ↗
w 6= ∅ (otherwise, set v = u/w)• if (u,w) not on outer face: • if (u,w) on outer face:
v
u
w• Add wN, wE, wS, and wW
![Page 113: Windrose Planarity - pub.informatik.uni-wuerzburg.de fileWindrose Planarity Embedding Graphs with Direction-Constrained Edges Published at SODA'16. Joint work with Patrizio Angelini,](https://reader031.vdocuments.site/reader031/viewer/2022022807/5cde865488c993680f8d29bb/html5/thumbnails/113.jpg)
Triangulated graphs
Let (G ,AQ) be a triangulated angular labeled graph.
v
u
w
• Assume that↗
u 6= ∅, ↗
w 6= ∅ (otherwise, set v = u/w)• if (u,w) not on outer face: • if (u,w) on outer face:
v
u
w• Add wN, wE, wS, and wW
![Page 114: Windrose Planarity - pub.informatik.uni-wuerzburg.de fileWindrose Planarity Embedding Graphs with Direction-Constrained Edges Published at SODA'16. Joint work with Patrizio Angelini,](https://reader031.vdocuments.site/reader031/viewer/2022022807/5cde865488c993680f8d29bb/html5/thumbnails/114.jpg)
Triangulated graphs
Let (G ,AQ) be a triangulated angular labeled graph.
v
u
w
• Assume that↗
u 6= ∅, ↗
w 6= ∅ (otherwise, set v = u/w)• if (u,w) not on outer face: • if (u,w) on outer face:
v
u
w• Add wN, wE, wS, and wW
![Page 115: Windrose Planarity - pub.informatik.uni-wuerzburg.de fileWindrose Planarity Embedding Graphs with Direction-Constrained Edges Published at SODA'16. Joint work with Patrizio Angelini,](https://reader031.vdocuments.site/reader031/viewer/2022022807/5cde865488c993680f8d29bb/html5/thumbnails/115.jpg)
Triangulated graphs
Let (G ,AQ) be a triangulated angular labeled graph.
v
u
w
• Assume that↗
u 6= ∅, ↗
w 6= ∅ (otherwise, set v = u/w)• if (u,w) not on outer face: • if (u,w) on outer face:
v
u
w• Add wN, wE, wS, and wW
![Page 116: Windrose Planarity - pub.informatik.uni-wuerzburg.de fileWindrose Planarity Embedding Graphs with Direction-Constrained Edges Published at SODA'16. Joint work with Patrizio Angelini,](https://reader031.vdocuments.site/reader031/viewer/2022022807/5cde865488c993680f8d29bb/html5/thumbnails/116.jpg)
Triangulated graphs
Let (G ,AQ) be a triangulated angular labeled graph.
v
u
w
• Assume that↗
u 6= ∅, ↗
w 6= ∅ (otherwise, set v = u/w)• if (u,w) not on outer face: • if (u,w) on outer face:
v
u
w• Add wN, wE, wS, and wW
![Page 117: Windrose Planarity - pub.informatik.uni-wuerzburg.de fileWindrose Planarity Embedding Graphs with Direction-Constrained Edges Published at SODA'16. Joint work with Patrizio Angelini,](https://reader031.vdocuments.site/reader031/viewer/2022022807/5cde865488c993680f8d29bb/html5/thumbnails/117.jpg)
Triangulated graphs
Let (G ,AQ) be a triangulated angular labeled graph.
v
u
w
• Assume that↗
u 6= ∅, ↗
w 6= ∅ (otherwise, set v = u/w)• if (u,w) not on outer face: • if (u,w) on outer face:
v
u
w• Add wN, wE, wS, and wW
![Page 118: Windrose Planarity - pub.informatik.uni-wuerzburg.de fileWindrose Planarity Embedding Graphs with Direction-Constrained Edges Published at SODA'16. Joint work with Patrizio Angelini,](https://reader031.vdocuments.site/reader031/viewer/2022022807/5cde865488c993680f8d29bb/html5/thumbnails/118.jpg)
Triangulated graphs
Let (G ,AQ) be a triangulated angular labeled graph.
v
u
w
• Assume that↗
u 6= ∅, ↗
w 6= ∅ (otherwise, set v = u/w)• if (u,w) not on outer face: • if (u,w) on outer face:
v
u
w• Add wN, wE, wS, and wW
![Page 119: Windrose Planarity - pub.informatik.uni-wuerzburg.de fileWindrose Planarity Embedding Graphs with Direction-Constrained Edges Published at SODA'16. Joint work with Patrizio Angelini,](https://reader031.vdocuments.site/reader031/viewer/2022022807/5cde865488c993680f8d29bb/html5/thumbnails/119.jpg)
Triangulated graphs
Let (G ,AQ) be a triangulated angular labeled graph.
v
u
w
• Assume that↗
u 6= ∅, ↗
w 6= ∅ (otherwise, set v = u/w)• if (u,w) not on outer face: • if (u,w) on outer face:
v
u
w• Add wN, wE, wS, and wW
![Page 120: Windrose Planarity - pub.informatik.uni-wuerzburg.de fileWindrose Planarity Embedding Graphs with Direction-Constrained Edges Published at SODA'16. Joint work with Patrizio Angelini,](https://reader031.vdocuments.site/reader031/viewer/2022022807/5cde865488c993680f8d29bb/html5/thumbnails/120.jpg)
Triangulated graphs
Let (G ,AQ) be a triangulated angular labeled graph.
v
u
w
• Assume that↗
u 6= ∅, ↗
w 6= ∅ (otherwise, set v = u/w)• if (u,w) not on outer face: • if (u,w) on outer face:
v
u
w• Add wN, wE, wS, and wW
![Page 121: Windrose Planarity - pub.informatik.uni-wuerzburg.de fileWindrose Planarity Embedding Graphs with Direction-Constrained Edges Published at SODA'16. Joint work with Patrizio Angelini,](https://reader031.vdocuments.site/reader031/viewer/2022022807/5cde865488c993680f8d29bb/html5/thumbnails/121.jpg)
Triangulated graphs
Let (G ,AQ) be a triangulated angular labeled graph.
v
u
w
• Assume that↗
u 6= ∅, ↗
w 6= ∅ (otherwise, set v = u/w)• if (u,w) not on outer face: • if (u,w) on outer face:
v
u
w• Add wN, wE, wS, and wW
![Page 122: Windrose Planarity - pub.informatik.uni-wuerzburg.de fileWindrose Planarity Embedding Graphs with Direction-Constrained Edges Published at SODA'16. Joint work with Patrizio Angelini,](https://reader031.vdocuments.site/reader031/viewer/2022022807/5cde865488c993680f8d29bb/html5/thumbnails/122.jpg)
Triangulated graphs
Let (G ,AQ) be a triangulated angular labeled graph.
v
u
w
• Assume that↗
u 6= ∅, ↗
w 6= ∅ (otherwise, set v = u/w)• if (u,w) not on outer face: • if (u,w) on outer face:
v
u
w• Add wN, wE, wS, and wW
Theorem.A triangulated q-constrained graph (G ,Q) is windrose planar⇔ AQ is angular→ draw with 1 bend per edge
on an O(n)× O(n) grid in O(n) time
![Page 123: Windrose Planarity - pub.informatik.uni-wuerzburg.de fileWindrose Planarity Embedding Graphs with Direction-Constrained Edges Published at SODA'16. Joint work with Patrizio Angelini,](https://reader031.vdocuments.site/reader031/viewer/2022022807/5cde865488c993680f8d29bb/html5/thumbnails/123.jpg)
Plane Graphs
Lemma.plane q-constrained graph (G ,Q)⇒ find a large-angle assignment L such that
AQ,L is angular (if it exists) in O(n log3 n) time
![Page 124: Windrose Planarity - pub.informatik.uni-wuerzburg.de fileWindrose Planarity Embedding Graphs with Direction-Constrained Edges Published at SODA'16. Joint work with Patrizio Angelini,](https://reader031.vdocuments.site/reader031/viewer/2022022807/5cde865488c993680f8d29bb/html5/thumbnails/124.jpg)
Plane Graphs
Lemma.plane q-constrained graph (G ,Q)⇒ find a large-angle assignment L such that
AQ,L is angular (if it exists) in O(n log3 n) time
Lemma.Plane angular labeled graph (G ,A)⇒ augment in O(n) time to a
triangulated labeled graph (G ′,A′)
![Page 125: Windrose Planarity - pub.informatik.uni-wuerzburg.de fileWindrose Planarity Embedding Graphs with Direction-Constrained Edges Published at SODA'16. Joint work with Patrizio Angelini,](https://reader031.vdocuments.site/reader031/viewer/2022022807/5cde865488c993680f8d29bb/html5/thumbnails/125.jpg)
Plane Graphs
Lemma.plane q-constrained graph (G ,Q)⇒ find a large-angle assignment L such that
AQ,L is angular (if it exists) in O(n log3 n) time
Lemma.Plane angular labeled graph (G ,A)⇒ augment in O(n) time to a
triangulated labeled graph (G ′,A′)
270◦
90◦
β
v1v2
v3
v4
C
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Plane Graphs
Lemma.plane q-constrained graph (G ,Q)⇒ find a large-angle assignment L such that
AQ,L is angular (if it exists) in O(n log3 n) time
Lemma.Plane angular labeled graph (G ,A)⇒ augment in O(n) time to a
triangulated labeled graph (G ′,A′)
180◦
90◦v1
v3
90◦ 0◦
C β
v2v4
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Plane Graphs
Lemma.plane q-constrained graph (G ,Q)⇒ find a large-angle assignment L such that
AQ,L is angular (if it exists) in O(n log3 n) time
Lemma.Plane angular labeled graph (G ,A)⇒ augment in O(n) time to a
triangulated labeled graph (G ′,A′)
Theorem.Plane q-constrained Graph⇒ test windrose planarity in O(n log3 n) time→ draw with 1 bend per edge on O(n)× O(n) grid
![Page 128: Windrose Planarity - pub.informatik.uni-wuerzburg.de fileWindrose Planarity Embedding Graphs with Direction-Constrained Edges Published at SODA'16. Joint work with Patrizio Angelini,](https://reader031.vdocuments.site/reader031/viewer/2022022807/5cde865488c993680f8d29bb/html5/thumbnails/128.jpg)
Further Results
Theorem.Windrose planar q-constrained graph (G ,Q) whose blocksare either edges or planar 3-trees⇒ straight-line windrose planar drawing
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Further Results
Theorem.Windrose planar q-constrained graph (G ,Q) whose blocksare either edges or planar 3-trees⇒ straight-line windrose planar drawing
Straight-line windrose planar drawingsrequire exponential area.
Theorem.
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Further Results
Theorem.Windrose planar q-constrained graph (G ,Q) whose blocksare either edges or planar 3-trees⇒ straight-line windrose planar drawing
Straight-line windrose planar drawingsrequire exponential area.
Theorem.
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Further Results
Theorem.Windrose planar q-constrained graph (G ,Q) whose blocksare either edges or planar 3-trees⇒ straight-line windrose planar drawing
Straight-line windrose planar drawingsrequire exponential area.
Theorem.
![Page 132: Windrose Planarity - pub.informatik.uni-wuerzburg.de fileWindrose Planarity Embedding Graphs with Direction-Constrained Edges Published at SODA'16. Joint work with Patrizio Angelini,](https://reader031.vdocuments.site/reader031/viewer/2022022807/5cde865488c993680f8d29bb/html5/thumbnails/132.jpg)
Further Results
Theorem.Windrose planar q-constrained graph (G ,Q) whose blocksare either edges or planar 3-trees⇒ straight-line windrose planar drawing
Straight-line windrose planar drawingsrequire exponential area.
Theorem.
![Page 133: Windrose Planarity - pub.informatik.uni-wuerzburg.de fileWindrose Planarity Embedding Graphs with Direction-Constrained Edges Published at SODA'16. Joint work with Patrizio Angelini,](https://reader031.vdocuments.site/reader031/viewer/2022022807/5cde865488c993680f8d29bb/html5/thumbnails/133.jpg)
Open problems
• Draw windrose planar graphs straight-line?
![Page 134: Windrose Planarity - pub.informatik.uni-wuerzburg.de fileWindrose Planarity Embedding Graphs with Direction-Constrained Edges Published at SODA'16. Joint work with Patrizio Angelini,](https://reader031.vdocuments.site/reader031/viewer/2022022807/5cde865488c993680f8d29bb/html5/thumbnails/134.jpg)
Open problems
• Draw windrose planar graphs straight-line?
• Generalizations: each edge has a set of possible directionsor allow more than two directions
![Page 135: Windrose Planarity - pub.informatik.uni-wuerzburg.de fileWindrose Planarity Embedding Graphs with Direction-Constrained Edges Published at SODA'16. Joint work with Patrizio Angelini,](https://reader031.vdocuments.site/reader031/viewer/2022022807/5cde865488c993680f8d29bb/html5/thumbnails/135.jpg)
Open problems
• Draw windrose planar graphs straight-line?
• Generalizations:
• What about bimonotone (relationship between each pairof vertices)? Not always straight-line!
each edge has a set of possible directionsor allow more than two directions
![Page 136: Windrose Planarity - pub.informatik.uni-wuerzburg.de fileWindrose Planarity Embedding Graphs with Direction-Constrained Edges Published at SODA'16. Joint work with Patrizio Angelini,](https://reader031.vdocuments.site/reader031/viewer/2022022807/5cde865488c993680f8d29bb/html5/thumbnails/136.jpg)
Open problems
• Draw windrose planar graphs straight-line?
• Generalizations:
• What about bimonotone (relationship between each pairof vertices)? Not always straight-line!
each edge has a set of possible directionsor allow more than two directions