geometry: combinatorics & algorithms eth zürich daniel...
TRANSCRIPT
![Page 1: Geometry: Combinatorics & Algorithms ETH Zürich Daniel Grafdgraf.ch/d/ETH/talks/Upward_Planarity_Daniel_Graf.pdf · Daniel Graf Checking Upward Planarity ETH Zürich NP-hard in general](https://reader036.vdocuments.site/reader036/viewer/2022071401/60eb31290dff3245823ee5e2/html5/thumbnails/1.jpg)
Daniel GrafETH ZürichGeometry: Combinatorics & Algorithms
Upw
ardPlanarity
![Page 2: Geometry: Combinatorics & Algorithms ETH Zürich Daniel Grafdgraf.ch/d/ETH/talks/Upward_Planarity_Daniel_Graf.pdf · Daniel Graf Checking Upward Planarity ETH Zürich NP-hard in general](https://reader036.vdocuments.site/reader036/viewer/2022071401/60eb31290dff3245823ee5e2/html5/thumbnails/2.jpg)
Daniel GrafETH ZürichDrawing Directed Graphs
1 2
3 4
![Page 3: Geometry: Combinatorics & Algorithms ETH Zürich Daniel Grafdgraf.ch/d/ETH/talks/Upward_Planarity_Daniel_Graf.pdf · Daniel Graf Checking Upward Planarity ETH Zürich NP-hard in general](https://reader036.vdocuments.site/reader036/viewer/2022071401/60eb31290dff3245823ee5e2/html5/thumbnails/3.jpg)
Daniel GrafETH ZürichDrawing Directed Graphs
1 2
3 4
(toposort)1
2
3
4
upward
![Page 4: Geometry: Combinatorics & Algorithms ETH Zürich Daniel Grafdgraf.ch/d/ETH/talks/Upward_Planarity_Daniel_Graf.pdf · Daniel Graf Checking Upward Planarity ETH Zürich NP-hard in general](https://reader036.vdocuments.site/reader036/viewer/2022071401/60eb31290dff3245823ee5e2/html5/thumbnails/4.jpg)
Daniel GrafETH ZürichDrawing Directed Graphs
1 2
3 4
(toposort)1
2
3
4
upward planar1 2
3 4
(Boyer Myrvold)
![Page 5: Geometry: Combinatorics & Algorithms ETH Zürich Daniel Grafdgraf.ch/d/ETH/talks/Upward_Planarity_Daniel_Graf.pdf · Daniel Graf Checking Upward Planarity ETH Zürich NP-hard in general](https://reader036.vdocuments.site/reader036/viewer/2022071401/60eb31290dff3245823ee5e2/html5/thumbnails/5.jpg)
Daniel GrafETH ZürichDrawing Directed Graphs
1 2
3 4
upward planar?
(toposort)1
2
3
4
upward planar1 2
3 4
(Boyer Myrvold)
![Page 6: Geometry: Combinatorics & Algorithms ETH Zürich Daniel Grafdgraf.ch/d/ETH/talks/Upward_Planarity_Daniel_Graf.pdf · Daniel Graf Checking Upward Planarity ETH Zürich NP-hard in general](https://reader036.vdocuments.site/reader036/viewer/2022071401/60eb31290dff3245823ee5e2/html5/thumbnails/6.jpg)
Daniel GrafETH ZürichDrawing Directed Graphs
1 2
3 4
1
2
3
4
upward planar?
(toposort)1
2
3
4
upward planar1 2
3 4
(Boyer Myrvold)
![Page 7: Geometry: Combinatorics & Algorithms ETH Zürich Daniel Grafdgraf.ch/d/ETH/talks/Upward_Planarity_Daniel_Graf.pdf · Daniel Graf Checking Upward Planarity ETH Zürich NP-hard in general](https://reader036.vdocuments.site/reader036/viewer/2022071401/60eb31290dff3245823ee5e2/html5/thumbnails/7.jpg)
Daniel GrafETH ZürichDrawing Directed Graphs
Acyclicity is not enough
![Page 8: Geometry: Combinatorics & Algorithms ETH Zürich Daniel Grafdgraf.ch/d/ETH/talks/Upward_Planarity_Daniel_Graf.pdf · Daniel Graf Checking Upward Planarity ETH Zürich NP-hard in general](https://reader036.vdocuments.site/reader036/viewer/2022071401/60eb31290dff3245823ee5e2/html5/thumbnails/8.jpg)
Daniel GrafETH ZürichChecking Upward Planarity
[GT95a] Ashim Garg and Roberto Tamassia. On the computational complexity of upward and rectilinear planarity testing. In Graph drawing, pages 286–297. Springer, 1995.[DBT88] Giuseppe Di Battista and Roberto Tamassia. Algorithms for plane representations of acyclic digraphs. Theoretical Computer Science, 61(2):175–198, 1988.[K87] David Kelly. Fundamentals of planar ordered sets. Discrete Mathematics, 63(2):197–216, 1987.[GT95b] Ashim Garg and Roberto Tamassia. Upward planarity testing. Order, 12(2):109–133, 1995.
![Page 9: Geometry: Combinatorics & Algorithms ETH Zürich Daniel Grafdgraf.ch/d/ETH/talks/Upward_Planarity_Daniel_Graf.pdf · Daniel Graf Checking Upward Planarity ETH Zürich NP-hard in general](https://reader036.vdocuments.site/reader036/viewer/2022071401/60eb31290dff3245823ee5e2/html5/thumbnails/9.jpg)
Daniel GrafETH ZürichChecking Upward Planarity
NP-hard in general [GT95a]
[GT95a] Ashim Garg and Roberto Tamassia. On the computational complexity of upward and rectilinear planarity testing. In Graph drawing, pages 286–297. Springer, 1995.[DBT88] Giuseppe Di Battista and Roberto Tamassia. Algorithms for plane representations of acyclic digraphs. Theoretical Computer Science, 61(2):175–198, 1988.[K87] David Kelly. Fundamentals of planar ordered sets. Discrete Mathematics, 63(2):197–216, 1987.[GT95b] Ashim Garg and Roberto Tamassia. Upward planarity testing. Order, 12(2):109–133, 1995.
![Page 10: Geometry: Combinatorics & Algorithms ETH Zürich Daniel Grafdgraf.ch/d/ETH/talks/Upward_Planarity_Daniel_Graf.pdf · Daniel Graf Checking Upward Planarity ETH Zürich NP-hard in general](https://reader036.vdocuments.site/reader036/viewer/2022071401/60eb31290dff3245823ee5e2/html5/thumbnails/10.jpg)
Daniel GrafETH ZürichChecking Upward Planarity
NP-hard in general [GT95a]but: nice characterization [DBT88],[K87]
[GT95a] Ashim Garg and Roberto Tamassia. On the computational complexity of upward and rectilinear planarity testing. In Graph drawing, pages 286–297. Springer, 1995.[DBT88] Giuseppe Di Battista and Roberto Tamassia. Algorithms for plane representations of acyclic digraphs. Theoretical Computer Science, 61(2):175–198, 1988.[K87] David Kelly. Fundamentals of planar ordered sets. Discrete Mathematics, 63(2):197–216, 1987.[GT95b] Ashim Garg and Roberto Tamassia. Upward planarity testing. Order, 12(2):109–133, 1995.
![Page 11: Geometry: Combinatorics & Algorithms ETH Zürich Daniel Grafdgraf.ch/d/ETH/talks/Upward_Planarity_Daniel_Graf.pdf · Daniel Graf Checking Upward Planarity ETH Zürich NP-hard in general](https://reader036.vdocuments.site/reader036/viewer/2022071401/60eb31290dff3245823ee5e2/html5/thumbnails/11.jpg)
Daniel GrafETH ZürichChecking Upward Planarity
NP-hard in general [GT95a]but: nice characterization [DBT88],[K87]
upward planar
[GT95a] Ashim Garg and Roberto Tamassia. On the computational complexity of upward and rectilinear planarity testing. In Graph drawing, pages 286–297. Springer, 1995.[DBT88] Giuseppe Di Battista and Roberto Tamassia. Algorithms for plane representations of acyclic digraphs. Theoretical Computer Science, 61(2):175–198, 1988.[K87] David Kelly. Fundamentals of planar ordered sets. Discrete Mathematics, 63(2):197–216, 1987.[GT95b] Ashim Garg and Roberto Tamassia. Upward planarity testing. Order, 12(2):109–133, 1995.
![Page 12: Geometry: Combinatorics & Algorithms ETH Zürich Daniel Grafdgraf.ch/d/ETH/talks/Upward_Planarity_Daniel_Graf.pdf · Daniel Graf Checking Upward Planarity ETH Zürich NP-hard in general](https://reader036.vdocuments.site/reader036/viewer/2022071401/60eb31290dff3245823ee5e2/html5/thumbnails/12.jpg)
Daniel GrafETH ZürichChecking Upward Planarity
NP-hard in general [GT95a]but: nice characterization [DBT88],[K87]
upward planar s-t-planar
[GT95a] Ashim Garg and Roberto Tamassia. On the computational complexity of upward and rectilinear planarity testing. In Graph drawing, pages 286–297. Springer, 1995.[DBT88] Giuseppe Di Battista and Roberto Tamassia. Algorithms for plane representations of acyclic digraphs. Theoretical Computer Science, 61(2):175–198, 1988.[K87] David Kelly. Fundamentals of planar ordered sets. Discrete Mathematics, 63(2):197–216, 1987.[GT95b] Ashim Garg and Roberto Tamassia. Upward planarity testing. Order, 12(2):109–133, 1995.
![Page 13: Geometry: Combinatorics & Algorithms ETH Zürich Daniel Grafdgraf.ch/d/ETH/talks/Upward_Planarity_Daniel_Graf.pdf · Daniel Graf Checking Upward Planarity ETH Zürich NP-hard in general](https://reader036.vdocuments.site/reader036/viewer/2022071401/60eb31290dff3245823ee5e2/html5/thumbnails/13.jpg)
Daniel GrafETH ZürichChecking Upward Planarity
NP-hard in general [GT95a]but: nice characterization [DBT88],[K87]
is spanningsubgraph of
upward planar s-t-planar
[GT95a] Ashim Garg and Roberto Tamassia. On the computational complexity of upward and rectilinear planarity testing. In Graph drawing, pages 286–297. Springer, 1995.[DBT88] Giuseppe Di Battista and Roberto Tamassia. Algorithms for plane representations of acyclic digraphs. Theoretical Computer Science, 61(2):175–198, 1988.[K87] David Kelly. Fundamentals of planar ordered sets. Discrete Mathematics, 63(2):197–216, 1987.[GT95b] Ashim Garg and Roberto Tamassia. Upward planarity testing. Order, 12(2):109–133, 1995.
![Page 14: Geometry: Combinatorics & Algorithms ETH Zürich Daniel Grafdgraf.ch/d/ETH/talks/Upward_Planarity_Daniel_Graf.pdf · Daniel Graf Checking Upward Planarity ETH Zürich NP-hard in general](https://reader036.vdocuments.site/reader036/viewer/2022071401/60eb31290dff3245823ee5e2/html5/thumbnails/14.jpg)
Daniel GrafETH ZürichChecking Upward Planarity
NP-hard in general [GT95a]but: nice characterization [DBT88],[K87]
is spanningsubgraph of
upward planar s-t-planar
[GT95a] Ashim Garg and Roberto Tamassia. On the computational complexity of upward and rectilinear planarity testing. In Graph drawing, pages 286–297. Springer, 1995.[DBT88] Giuseppe Di Battista and Roberto Tamassia. Algorithms for plane representations of acyclic digraphs. Theoretical Computer Science, 61(2):175–198, 1988.[K87] David Kelly. Fundamentals of planar ordered sets. Discrete Mathematics, 63(2):197–216, 1987.[GT95b] Ashim Garg and Roberto Tamassia. Upward planarity testing. Order, 12(2):109–133, 1995.
![Page 15: Geometry: Combinatorics & Algorithms ETH Zürich Daniel Grafdgraf.ch/d/ETH/talks/Upward_Planarity_Daniel_Graf.pdf · Daniel Graf Checking Upward Planarity ETH Zürich NP-hard in general](https://reader036.vdocuments.site/reader036/viewer/2022071401/60eb31290dff3245823ee5e2/html5/thumbnails/15.jpg)
Daniel GrafETH ZürichChecking Upward Planarity
NP-hard in general [GT95a]but: nice characterization [DBT88],[K87]
is spanningsubgraph of
upward planar s-t-planar
[GT95a] Ashim Garg and Roberto Tamassia. On the computational complexity of upward and rectilinear planarity testing. In Graph drawing, pages 286–297. Springer, 1995.[DBT88] Giuseppe Di Battista and Roberto Tamassia. Algorithms for plane representations of acyclic digraphs. Theoretical Computer Science, 61(2):175–198, 1988.[K87] David Kelly. Fundamentals of planar ordered sets. Discrete Mathematics, 63(2):197–216, 1987.[GT95b] Ashim Garg and Roberto Tamassia. Upward planarity testing. Order, 12(2):109–133, 1995.
⇒ NP-complete [GT95b]
![Page 16: Geometry: Combinatorics & Algorithms ETH Zürich Daniel Grafdgraf.ch/d/ETH/talks/Upward_Planarity_Daniel_Graf.pdf · Daniel Graf Checking Upward Planarity ETH Zürich NP-hard in general](https://reader036.vdocuments.site/reader036/viewer/2022071401/60eb31290dff3245823ee5e2/html5/thumbnails/16.jpg)
Daniel GrafETH ZürichChecking Upward Planarity
Special cases in P:
[DBT88] Giuseppe Di Battista and Roberto Tamassia. Algorithms for plane representations of acyclic digraphs. Theoretical Computer Science, 61(2):175–198, 1988.
![Page 17: Geometry: Combinatorics & Algorithms ETH Zürich Daniel Grafdgraf.ch/d/ETH/talks/Upward_Planarity_Daniel_Graf.pdf · Daniel Graf Checking Upward Planarity ETH Zürich NP-hard in general](https://reader036.vdocuments.site/reader036/viewer/2022071401/60eb31290dff3245823ee5e2/html5/thumbnails/17.jpg)
Daniel GrafETH ZürichChecking Upward Planarity
Special cases in P:• single source, single sink [DBT88]
[DBT88] Giuseppe Di Battista and Roberto Tamassia. Algorithms for plane representations of acyclic digraphs. Theoretical Computer Science, 61(2):175–198, 1988.
![Page 18: Geometry: Combinatorics & Algorithms ETH Zürich Daniel Grafdgraf.ch/d/ETH/talks/Upward_Planarity_Daniel_Graf.pdf · Daniel Graf Checking Upward Planarity ETH Zürich NP-hard in general](https://reader036.vdocuments.site/reader036/viewer/2022071401/60eb31290dff3245823ee5e2/html5/thumbnails/18.jpg)
Daniel GrafETH ZürichChecking Upward Planarity
Special cases in P:• single source, single sink [DBT88]
[DBT88] Giuseppe Di Battista and Roberto Tamassia. Algorithms for plane representations of acyclic digraphs. Theoretical Computer Science, 61(2):175–198, 1988.
![Page 19: Geometry: Combinatorics & Algorithms ETH Zürich Daniel Grafdgraf.ch/d/ETH/talks/Upward_Planarity_Daniel_Graf.pdf · Daniel Graf Checking Upward Planarity ETH Zürich NP-hard in general](https://reader036.vdocuments.site/reader036/viewer/2022071401/60eb31290dff3245823ee5e2/html5/thumbnails/19.jpg)
Daniel GrafETH ZürichChecking Upward Planarity
Special cases in P:• single source, single sink [DBT88]
planar?
[DBT88] Giuseppe Di Battista and Roberto Tamassia. Algorithms for plane representations of acyclic digraphs. Theoretical Computer Science, 61(2):175–198, 1988.
![Page 20: Geometry: Combinatorics & Algorithms ETH Zürich Daniel Grafdgraf.ch/d/ETH/talks/Upward_Planarity_Daniel_Graf.pdf · Daniel Graf Checking Upward Planarity ETH Zürich NP-hard in general](https://reader036.vdocuments.site/reader036/viewer/2022071401/60eb31290dff3245823ee5e2/html5/thumbnails/20.jpg)
Daniel GrafETH ZürichChecking Upward Planarity
Special cases in P:• fixed embedding [BDB91]⇒ maximal planar graphs
[BDB91] Paola Bertolazzi and Giuseppe Di Battista. On upward drawing testing of triconnected digraphs. In Proceedings of the seventh annual symposium on computational geometry, pages 272–280. ACM, 1991.
1
2
3
4
1
2
3
4
![Page 21: Geometry: Combinatorics & Algorithms ETH Zürich Daniel Grafdgraf.ch/d/ETH/talks/Upward_Planarity_Daniel_Graf.pdf · Daniel Graf Checking Upward Planarity ETH Zürich NP-hard in general](https://reader036.vdocuments.site/reader036/viewer/2022071401/60eb31290dff3245823ee5e2/html5/thumbnails/21.jpg)
Daniel GrafETH ZürichStraight Line Upward Drawing
1
2
3
4
1
2
3
4
[DBTT92] Giuseppe Di Battista, Roberto Tamassia, and Ioannis G Tollis. Area requirement and symmetry display of planar upward drawings. Discrete & Computational Geometry, 7(1):381– 401, 1992.
![Page 22: Geometry: Combinatorics & Algorithms ETH Zürich Daniel Grafdgraf.ch/d/ETH/talks/Upward_Planarity_Daniel_Graf.pdf · Daniel Graf Checking Upward Planarity ETH Zürich NP-hard in general](https://reader036.vdocuments.site/reader036/viewer/2022071401/60eb31290dff3245823ee5e2/html5/thumbnails/22.jpg)
Daniel GrafETH ZürichStraight Line Upward Drawing
Always possible, but might need large grid [DBTT92]
1
2
3
4
1
2
3
4
[DBTT92] Giuseppe Di Battista, Roberto Tamassia, and Ioannis G Tollis. Area requirement and symmetry display of planar upward drawings. Discrete & Computational Geometry, 7(1):381– 401, 1992.
![Page 23: Geometry: Combinatorics & Algorithms ETH Zürich Daniel Grafdgraf.ch/d/ETH/talks/Upward_Planarity_Daniel_Graf.pdf · Daniel Graf Checking Upward Planarity ETH Zürich NP-hard in general](https://reader036.vdocuments.site/reader036/viewer/2022071401/60eb31290dff3245823ee5e2/html5/thumbnails/23.jpg)
Daniel GrafETH ZürichUpward Planar Orientations
[FGW13] Fabrizio Frati, Joachim Gudmundsson, and Emo Welzl. On the number of upward planar orientations of maximal planar graphs. Theoretical Computer Science, 544:32–59, 2014.
![Page 24: Geometry: Combinatorics & Algorithms ETH Zürich Daniel Grafdgraf.ch/d/ETH/talks/Upward_Planarity_Daniel_Graf.pdf · Daniel Graf Checking Upward Planarity ETH Zürich NP-hard in general](https://reader036.vdocuments.site/reader036/viewer/2022071401/60eb31290dff3245823ee5e2/html5/thumbnails/24.jpg)
Daniel GrafETH ZürichUpward Planar Orientations
[FGW13] Fabrizio Frati, Joachim Gudmundsson, and Emo Welzl. On the number of upward planar orientations of maximal planar graphs. Theoretical Computer Science, 544:32–59, 2014.
![Page 25: Geometry: Combinatorics & Algorithms ETH Zürich Daniel Grafdgraf.ch/d/ETH/talks/Upward_Planarity_Daniel_Graf.pdf · Daniel Graf Checking Upward Planarity ETH Zürich NP-hard in general](https://reader036.vdocuments.site/reader036/viewer/2022071401/60eb31290dff3245823ee5e2/html5/thumbnails/25.jpg)
Daniel GrafETH ZürichUpward Planar Orientations
[FGW13] Fabrizio Frati, Joachim Gudmundsson, and Emo Welzl. On the number of upward planar orientations of maximal planar graphs. Theoretical Computer Science, 544:32–59, 2014.
![Page 26: Geometry: Combinatorics & Algorithms ETH Zürich Daniel Grafdgraf.ch/d/ETH/talks/Upward_Planarity_Daniel_Graf.pdf · Daniel Graf Checking Upward Planarity ETH Zürich NP-hard in general](https://reader036.vdocuments.site/reader036/viewer/2022071401/60eb31290dff3245823ee5e2/html5/thumbnails/26.jpg)
Daniel GrafETH ZürichUpward Planar Orientations
[FGW13] Fabrizio Frati, Joachim Gudmundsson, and Emo Welzl. On the number of upward planar orientations of maximal planar graphs. Theoretical Computer Science, 544:32–59, 2014.
![Page 27: Geometry: Combinatorics & Algorithms ETH Zürich Daniel Grafdgraf.ch/d/ETH/talks/Upward_Planarity_Daniel_Graf.pdf · Daniel Graf Checking Upward Planarity ETH Zürich NP-hard in general](https://reader036.vdocuments.site/reader036/viewer/2022071401/60eb31290dff3245823ee5e2/html5/thumbnails/27.jpg)
Daniel GrafETH ZürichUpward Planar Orientations
X
[FGW13] Fabrizio Frati, Joachim Gudmundsson, and Emo Welzl. On the number of upward planar orientations of maximal planar graphs. Theoretical Computer Science, 544:32–59, 2014.
![Page 28: Geometry: Combinatorics & Algorithms ETH Zürich Daniel Grafdgraf.ch/d/ETH/talks/Upward_Planarity_Daniel_Graf.pdf · Daniel Graf Checking Upward Planarity ETH Zürich NP-hard in general](https://reader036.vdocuments.site/reader036/viewer/2022071401/60eb31290dff3245823ee5e2/html5/thumbnails/28.jpg)
Daniel GrafETH ZürichUpward Planar Orientations
XHow many?
[FGW13] Fabrizio Frati, Joachim Gudmundsson, and Emo Welzl. On the number of upward planar orientations of maximal planar graphs. Theoretical Computer Science, 544:32–59, 2014.
![Page 29: Geometry: Combinatorics & Algorithms ETH Zürich Daniel Grafdgraf.ch/d/ETH/talks/Upward_Planarity_Daniel_Graf.pdf · Daniel Graf Checking Upward Planarity ETH Zürich NP-hard in general](https://reader036.vdocuments.site/reader036/viewer/2022071401/60eb31290dff3245823ee5e2/html5/thumbnails/29.jpg)
Daniel GrafETH ZürichUpward Planar Orientations
Maximal planar: some graphs with orientations[FGW13] and some graphs with orientations
O(2n)Ω(2.5n)
XHow many?
[FGW13] Fabrizio Frati, Joachim Gudmundsson, and Emo Welzl. On the number of upward planar orientations of maximal planar graphs. Theoretical Computer Science, 544:32–59, 2014.
![Page 30: Geometry: Combinatorics & Algorithms ETH Zürich Daniel Grafdgraf.ch/d/ETH/talks/Upward_Planarity_Daniel_Graf.pdf · Daniel Graf Checking Upward Planarity ETH Zürich NP-hard in general](https://reader036.vdocuments.site/reader036/viewer/2022071401/60eb31290dff3245823ee5e2/html5/thumbnails/30.jpg)
Daniel GrafETH ZürichUpward Planar Orientations
Maximal planar: some graphs with orientations[FGW13] and some graphs with orientations
O(2n)Ω(2.5n) Rest: open
XHow many?
[FGW13] Fabrizio Frati, Joachim Gudmundsson, and Emo Welzl. On the number of upward planar orientations of maximal planar graphs. Theoretical Computer Science, 544:32–59, 2014.
![Page 31: Geometry: Combinatorics & Algorithms ETH Zürich Daniel Grafdgraf.ch/d/ETH/talks/Upward_Planarity_Daniel_Graf.pdf · Daniel Graf Checking Upward Planarity ETH Zürich NP-hard in general](https://reader036.vdocuments.site/reader036/viewer/2022071401/60eb31290dff3245823ee5e2/html5/thumbnails/31.jpg)
Daniel GrafETH ZürichChecking Variations
[BDBD98] Paola Bertolazzi, Giuseppe Di Battista, and Walter Didimo. Quasi-upward planarity. In Graph Drawing, pages 15–29. Springer, 1998.[FKPTW13] Fabrizio Frati, Michael Kaufmann, Janos Pach, Csaba D Toth, and David R Wood. On the upward planarity of mixed plane graphs. In Graph Drawing, pages 1–12. Springer, 2013.[BDP14] Carla Binucci, Walter Didimo, and Maurizio Patrignani. Upward and quasi-upward pla- narity testing of embedded mixed graphs. Theoretical Computer Science, 526:75–89, 2014.
![Page 32: Geometry: Combinatorics & Algorithms ETH Zürich Daniel Grafdgraf.ch/d/ETH/talks/Upward_Planarity_Daniel_Graf.pdf · Daniel Graf Checking Upward Planarity ETH Zürich NP-hard in general](https://reader036.vdocuments.site/reader036/viewer/2022071401/60eb31290dff3245823ee5e2/html5/thumbnails/32.jpg)
Daniel GrafETH ZürichChecking Variations
quasi upward
[BDBD98] Paola Bertolazzi, Giuseppe Di Battista, and Walter Didimo. Quasi-upward planarity. In Graph Drawing, pages 15–29. Springer, 1998.[FKPTW13] Fabrizio Frati, Michael Kaufmann, Janos Pach, Csaba D Toth, and David R Wood. On the upward planarity of mixed plane graphs. In Graph Drawing, pages 1–12. Springer, 2013.[BDP14] Carla Binucci, Walter Didimo, and Maurizio Patrignani. Upward and quasi-upward pla- narity testing of embedded mixed graphs. Theoretical Computer Science, 526:75–89, 2014.
![Page 33: Geometry: Combinatorics & Algorithms ETH Zürich Daniel Grafdgraf.ch/d/ETH/talks/Upward_Planarity_Daniel_Graf.pdf · Daniel Graf Checking Upward Planarity ETH Zürich NP-hard in general](https://reader036.vdocuments.site/reader036/viewer/2022071401/60eb31290dff3245823ee5e2/html5/thumbnails/33.jpg)
Daniel GrafETH ZürichChecking Variations
[BDBD98] given ɸ, in P
quasi upward
[BDBD98] Paola Bertolazzi, Giuseppe Di Battista, and Walter Didimo. Quasi-upward planarity. In Graph Drawing, pages 15–29. Springer, 1998.[FKPTW13] Fabrizio Frati, Michael Kaufmann, Janos Pach, Csaba D Toth, and David R Wood. On the upward planarity of mixed plane graphs. In Graph Drawing, pages 1–12. Springer, 2013.[BDP14] Carla Binucci, Walter Didimo, and Maurizio Patrignani. Upward and quasi-upward pla- narity testing of embedded mixed graphs. Theoretical Computer Science, 526:75–89, 2014.
![Page 34: Geometry: Combinatorics & Algorithms ETH Zürich Daniel Grafdgraf.ch/d/ETH/talks/Upward_Planarity_Daniel_Graf.pdf · Daniel Graf Checking Upward Planarity ETH Zürich NP-hard in general](https://reader036.vdocuments.site/reader036/viewer/2022071401/60eb31290dff3245823ee5e2/html5/thumbnails/34.jpg)
Daniel GrafETH ZürichChecking Variations
[BDBD98] given ɸ, in P
quasi upward mixed graphs
[BDBD98] Paola Bertolazzi, Giuseppe Di Battista, and Walter Didimo. Quasi-upward planarity. In Graph Drawing, pages 15–29. Springer, 1998.[FKPTW13] Fabrizio Frati, Michael Kaufmann, Janos Pach, Csaba D Toth, and David R Wood. On the upward planarity of mixed plane graphs. In Graph Drawing, pages 1–12. Springer, 2013.[BDP14] Carla Binucci, Walter Didimo, and Maurizio Patrignani. Upward and quasi-upward pla- narity testing of embedded mixed graphs. Theoretical Computer Science, 526:75–89, 2014.
![Page 35: Geometry: Combinatorics & Algorithms ETH Zürich Daniel Grafdgraf.ch/d/ETH/talks/Upward_Planarity_Daniel_Graf.pdf · Daniel Graf Checking Upward Planarity ETH Zürich NP-hard in general](https://reader036.vdocuments.site/reader036/viewer/2022071401/60eb31290dff3245823ee5e2/html5/thumbnails/35.jpg)
Daniel GrafETH ZürichChecking Variations
[BDBD98] given ɸ, in P [FKPTW13] some classes in P
quasi upward mixed graphs
[BDBD98] Paola Bertolazzi, Giuseppe Di Battista, and Walter Didimo. Quasi-upward planarity. In Graph Drawing, pages 15–29. Springer, 1998.[FKPTW13] Fabrizio Frati, Michael Kaufmann, Janos Pach, Csaba D Toth, and David R Wood. On the upward planarity of mixed plane graphs. In Graph Drawing, pages 1–12. Springer, 2013.[BDP14] Carla Binucci, Walter Didimo, and Maurizio Patrignani. Upward and quasi-upward pla- narity testing of embedded mixed graphs. Theoretical Computer Science, 526:75–89, 2014.
![Page 36: Geometry: Combinatorics & Algorithms ETH Zürich Daniel Grafdgraf.ch/d/ETH/talks/Upward_Planarity_Daniel_Graf.pdf · Daniel Graf Checking Upward Planarity ETH Zürich NP-hard in general](https://reader036.vdocuments.site/reader036/viewer/2022071401/60eb31290dff3245823ee5e2/html5/thumbnails/36.jpg)
Daniel GrafETH ZürichChecking Variations
[BDBD98] given ɸ, in P [FKPTW13] some classes in P
Open: mixed but fixed ɸ?
quasi upward mixed graphs
[BDBD98] Paola Bertolazzi, Giuseppe Di Battista, and Walter Didimo. Quasi-upward planarity. In Graph Drawing, pages 15–29. Springer, 1998.[FKPTW13] Fabrizio Frati, Michael Kaufmann, Janos Pach, Csaba D Toth, and David R Wood. On the upward planarity of mixed plane graphs. In Graph Drawing, pages 1–12. Springer, 2013.[BDP14] Carla Binucci, Walter Didimo, and Maurizio Patrignani. Upward and quasi-upward pla- narity testing of embedded mixed graphs. Theoretical Computer Science, 526:75–89, 2014.
![Page 37: Geometry: Combinatorics & Algorithms ETH Zürich Daniel Grafdgraf.ch/d/ETH/talks/Upward_Planarity_Daniel_Graf.pdf · Daniel Graf Checking Upward Planarity ETH Zürich NP-hard in general](https://reader036.vdocuments.site/reader036/viewer/2022071401/60eb31290dff3245823ee5e2/html5/thumbnails/37.jpg)
Daniel GrafETH ZürichChecking Variations
[BDBD98] given ɸ, in P [FKPTW13] some classes in P
Open: mixed but fixed ɸ?- mixed → in P
quasi upward mixed graphs
[BDBD98] Paola Bertolazzi, Giuseppe Di Battista, and Walter Didimo. Quasi-upward planarity. In Graph Drawing, pages 15–29. Springer, 1998.[FKPTW13] Fabrizio Frati, Michael Kaufmann, Janos Pach, Csaba D Toth, and David R Wood. On the upward planarity of mixed plane graphs. In Graph Drawing, pages 1–12. Springer, 2013.[BDP14] Carla Binucci, Walter Didimo, and Maurizio Patrignani. Upward and quasi-upward pla- narity testing of embedded mixed graphs. Theoretical Computer Science, 526:75–89, 2014.
![Page 38: Geometry: Combinatorics & Algorithms ETH Zürich Daniel Grafdgraf.ch/d/ETH/talks/Upward_Planarity_Daniel_Graf.pdf · Daniel Graf Checking Upward Planarity ETH Zürich NP-hard in general](https://reader036.vdocuments.site/reader036/viewer/2022071401/60eb31290dff3245823ee5e2/html5/thumbnails/38.jpg)
Daniel GrafETH ZürichChecking Variations
[BDBD98] given ɸ, in P [FKPTW13] some classes in P
Open: mixed but fixed ɸ?+ quasi → NP-hard [BDP14]- mixed → in P
quasi upward mixed graphs
[BDBD98] Paola Bertolazzi, Giuseppe Di Battista, and Walter Didimo. Quasi-upward planarity. In Graph Drawing, pages 15–29. Springer, 1998.[FKPTW13] Fabrizio Frati, Michael Kaufmann, Janos Pach, Csaba D Toth, and David R Wood. On the upward planarity of mixed plane graphs. In Graph Drawing, pages 1–12. Springer, 2013.[BDP14] Carla Binucci, Walter Didimo, and Maurizio Patrignani. Upward and quasi-upward pla- narity testing of embedded mixed graphs. Theoretical Computer Science, 526:75–89, 2014.