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Daniel GrafETH ZürichGeometry: Combinatorics & Algorithms
Upw
ardPlanarity
Daniel GrafETH ZürichDrawing Directed Graphs
1 2
3 4
Daniel GrafETH ZürichDrawing Directed Graphs
1 2
3 4
(toposort)1
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4
upward
Daniel GrafETH ZürichDrawing Directed Graphs
1 2
3 4
(toposort)1
2
3
4
upward planar1 2
3 4
(Boyer Myrvold)
Daniel GrafETH ZürichDrawing Directed Graphs
1 2
3 4
upward planar?
(toposort)1
2
3
4
upward planar1 2
3 4
(Boyer Myrvold)
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)
Daniel GrafETH ZürichDrawing Directed Graphs
Acyclicity is not enough
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.
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.
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.
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.
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.
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.
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.
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]
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.
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.
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.
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.
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.
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Daniel GrafETH ZürichStraight Line Upward Drawing
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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.
Daniel GrafETH ZürichStraight Line Upward Drawing
Always possible, but might need large grid [DBTT92]
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1
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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.
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.
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.
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.
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.
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.
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.
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.
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.
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.
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.
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.
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.
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.
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.
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.
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.
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