Upward Planar Morphs
We prove that, given two topologically-equivalent upward planar straight-line drawings of an n -vertex directed graph G , there always exists a morph between them such that all the intermediate drawings of the morph are upward planar and straight-line. Such a morph consists of O (1) morphing steps i...
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| Published in: | Algorithmica Vol. 82; no. 10; pp. 2985 - 3017 |
|---|---|
| Main Authors: | , , , , |
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| Language: | English |
| Published: |
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01.10.2020
Springer Nature B.V |
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| ISSN: | 0178-4617, 1432-0541 |
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| Abstract | We prove that, given two topologically-equivalent upward planar straight-line drawings of an
n
-vertex directed graph
G
, there always exists a morph between them such that all the intermediate drawings of the morph are upward planar and straight-line. Such a morph consists of
O
(1) morphing steps if
G
is a reduced planar
st
-graph,
O
(
n
) morphing steps if
G
is a planar
st
-graph,
O
(
n
) morphing steps if
G
is a reduced upward planar graph, and
O
(
n
2
)
morphing steps if
G
is a general upward planar graph. Further, we show that
Ω
(
n
)
morphing steps might be necessary for an upward planar morph between two topologically-equivalent upward planar straight-line drawings of an
n
-vertex path. |
|---|---|
| AbstractList | We prove that, given two topologically-equivalent upward planar straight-line drawings of an
n
-vertex directed graph
G
, there always exists a morph between them such that all the intermediate drawings of the morph are upward planar and straight-line. Such a morph consists of
O
(1) morphing steps if
G
is a reduced planar
st
-graph,
O
(
n
) morphing steps if
G
is a planar
st
-graph,
O
(
n
) morphing steps if
G
is a reduced upward planar graph, and
O
(
n
2
)
morphing steps if
G
is a general upward planar graph. Further, we show that
Ω
(
n
)
morphing steps might be necessary for an upward planar morph between two topologically-equivalent upward planar straight-line drawings of an
n
-vertex path. We prove that, given two topologically-equivalent upward planar straight-line drawings of an n-vertex directed graph G, there always exists a morph between them such that all the intermediate drawings of the morph are upward planar and straight-line. Such a morph consists of O(1) morphing steps if G is a reduced planar st-graph, O(n) morphing steps if G is a planar st-graph, O(n) morphing steps if G is a reduced upward planar graph, and O(n2) morphing steps if G is a general upward planar graph. Further, we show that Ω(n) morphing steps might be necessary for an upward planar morph between two topologically-equivalent upward planar straight-line drawings of an n-vertex path. |
| Author | Da Lozzo, Giordano Frati, Fabrizio Patrignani, Maurizio Roselli, Vincenzo Di Battista, Giuseppe |
| Author_xml | – sequence: 1 givenname: Giordano surname: Da Lozzo fullname: Da Lozzo, Giordano organization: Roma Tre University – sequence: 2 givenname: Giuseppe surname: Di Battista fullname: Di Battista, Giuseppe organization: Roma Tre University – sequence: 3 givenname: Fabrizio orcidid: 0000-0001-5987-8713 surname: Frati fullname: Frati, Fabrizio email: frati@dia.uniroma3.it organization: Roma Tre University – sequence: 4 givenname: Maurizio surname: Patrignani fullname: Patrignani, Maurizio organization: Roma Tre University – sequence: 5 givenname: Vincenzo surname: Roselli fullname: Roselli, Vincenzo organization: Roma Tre University |
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| Cites_doi | 10.1007/BF02187706 10.1007/BF02187850 10.1007/11618058_2 10.1142/S0218195994000215 10.1137/S0097539794277123 10.1016/j.comgeo.2019.07.007 10.1007/BF03015194 10.1137/16M1069171 10.1007/BF02187705 10.1007/978-3-030-24766-9_25 10.1007/3-540-16078-7_71 10.1007/BF01188716 10.1137/S0097539794279626 10.2307/1967770 10.1080/00029890.1944.11999082 10.1016/j.tcs.2014.04.024 10.1016/j.jda.2009.05.003 10.1145/321850.321852 10.1007/978-3-030-24766-9_5 10.1137/S0097539792235724 10.1007/978-3-319-03841-4_5 10.1016/0304-3975(88)90123-5 10.1137/1.9781611973105.119 10.1007/978-3-030-04414-5_7 10.1016/0095-8956(83)90038-2 |
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| Issue | 10 |
| Keywords | Planar morph Graph drawing Upward planarity Directed graph |
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| References | HopcroftJTarjanREfficient planarity testingJ. ACM197421454956835938710.1145/321850.321852 Tamassia, R., Tollis, I.G.: Algorithms for visibility representations of planar graphs. In: Monien, B., Vidal-Naquet, G. (eds.) STACS 86, 3rd Annual Symposium on Theoretical Aspects of Computer Science, Orsay, France, 16–18 January 1986, Proceedings, volume 210 of Lecture Notes in Computer Science, pp. 130–141. Springer (1986) Barrera-Cruz, F., Haxell, P., Lubiw, A.: Morphing planar graph drawings with unidirectional moves. In: Mexican Conference on Discrete Mathematics and Computational Geometry, pp. 57–65 (2013). arXiv:1411.6185 RosenstiehlPTarjanRERectilinear planar layouts and bipolar orientations of planar graphsDiscrete Comput. Geom.1986134335386636910.1007/BF02187706 Roselli, V.: Morphing and visiting drawings of graphs. Ph.D. thesis, Università degli Studi di Roma “Roma Tre”, Dottorato di Ricerca in Ingegneria, Sezione Informatica ed Automazione, XXVI Ciclo (2014) Da Lozzo, G., Di Battista, G., Frati, F., Patrignani, M., Roselli, V.: Upward planar morphs. In: Biedl, T., Kerren, A. (eds.) 26th International Symposium on Graph Drawing and Network Visualization (GD ’18), Lecture Notes in Computer Science. Springer (2018) (to appear) Di BattistaGTamassiaRAlgorithms for plane representations of acyclic digraphsTheor. Comput. Sci.19886117519898024110.1016/0304-3975(88)90123-5 Angelini, P., Frati, F., Patrignani, M., Roselli, V.: Morphing planar graph drawings efficiently. In: Wismath, S., Wolff, A. (eds.) Proceedings of 21st International Symposium on Graph Drawing (GD ’13), Lecture Notes in Computer Science. Springer (2013) AlamdariSAngeliniPBarrera-CruzFChanTMDa LozzoGDi BattistaGFratiFHaxellPLubiwAPatrignaniMRoselliVSinglaSWilkinsonBTHow to morph planar graph drawingsSIAM J. Comput.2017462824852364063210.1137/16M1069171 MehlhornKData Structures and Algorithms: Multi-dimensional Searching and Computational Geometry1984BerlinSpringer0556.68003 BertolazziPDi BattistaGLiottaGManninoCUpward drawings of triconnected digraphsAlgorithmica1994126476497129781010.1007/BF01188716 Di BattistaGEadesPTamassiaRTollisIGGraph Drawing: Algorithms for the Visualization of Graphs1999Upper Saddle RiverPrentice-Hall1057.68653 GargATamassiaROn the computational complexity of upward and rectilinear planarity testingSIAM J. Comput.2002312601625186129210.1137/S0097539794277123 Biedl, T.C., Lubiw, A., Spriggs, M.J.: Morphing planar graphs while preserving edge directions. In: Healy, P., Nikolov, N.S. (eds.) 13th International Symposium on Graph Drawing (GD ’05), volume 3843 of Lecture Notes in Computer Science, pp. 13–24. Springer (2006) Aichholzer, O., Aloupis, G., Demaine, E.D., Demaine, M.L., Dujmovic, V., Hurtado, F., Lubiw, A., Rote, G., Schulz, A., Souvaine, D.L., Winslow, A.: Convexifying polygons without losing visibilities. In: 23rd Canadian Conference on Computational Geometry (CCCG ’11) (2011) CairnsSSDeformations of plane rectilinear complexesAm. Math. Mon.19445152472521027310.1080/00029890.1944.11999082 ThomassenCDeformations of plane graphsJ. Combin. Theory Ser. B198334324425771444810.1016/0095-8956(83)90038-2 BertolazziPCohenRFDi BattistaGTamassiaRTollisIGHow to draw a series-parallel digraphInt. J. Comput. Geom. Appl.199444385402131091110.1142/S0218195994000215 Schnyder, W.: Embedding planar graphs on the grid. In: Proceedings of the First Annual ACM-SIAM Symposium on Discrete Algorithms, SODA ’90, pp. 138–148, Philadelphia, PA, USA. Society for Industrial and Applied Mathematics (1990) AngeliniPDa LozzoGDi BattistaGFratiFPatrignaniMRoselliVEsparzaJFraigniaudPHusfeldtTKoutsoupiasEMorphing planar graph drawings optimally41st InternationalColloquium on Automata, Languages and Programming (ICALP ’14), volume of 8572 Lecture Notes in Computer Science2014BerlinSpringer126137 BertolazziPDi BattistaGManninoCTamassiaROptimal upward planarity testing of single-source digraphsSIAM J. Comput.1998271132169161482110.1137/S0097539794279626 KleistLKlemzBLubiwASchlipfLStaalsFStrashDConvexity-increasing morphs of planar graphsComput. Geom.2019846988399770510.1016/j.comgeo.2019.07.007 Di BattistaGTamassiaRTollisIGArea requirement and symmetry display of planar upward drawingsDiscrete Comput. Geom.199274381401114895310.1007/BF02187850 SmithHLOn continuous representations of a square upon itselfAnn. Math.1917192137141150252210.2307/1967770 Da Lozzo, G., Di Battista, G., Frati, F.: Extending upward planar graph drawings. In: WADS, volume 11646 of Lecture Notes in Computer Science, pp. 339–352. Springer (2019) Barrera-Cruz, F., Borrazzo, M., Da Lozzo, G., Di Battista, G., Frati, F., Patrignani, M., Roselli, V.: How to morph a tree on a small grid. In: Friggstad, Z., Sack, J., Salavatipour, M.R. (eds.) Algorithms and Data Structures: 16th International Symposium, WADS 2019, Edmonton, AB, Canada, 5–7 August 2019, Proceedings, volume 11646 of Lecture Notes in Computer Science, pp. 57–70. Springer, Berlin (2019) CohenRFDi BattistaGTamassiaRTollisIGDynamic graph drawings: trees, series-parallel digraphs, and planar ST-digraphsSIAM J. Comput.19952459701001135075410.1137/S0097539792235724 van Goethem, A., Verbeek, K.: Optimal morphs of planar orthogonal drawings. In: 34th International Symposium on Computational Geometry, SoCG 2018, 11–14 June 2018, Budapest, Hungary, pp. 42:1–42:14 (2018) TamassiaRTollisIGA unified approach to visibility representations of planar graphsDiscrete Comput. Geom.1986132134186636810.1007/BF02187705 TietzeHÜber stetige abbildungen einer quadratfläche auf sich selbstRendiconti del Circolo Matematico di Palermo191438124730410.1007/BF03015194 FáryIOn straight-line representation of planar graphsActa Scientiarum Mathematicarum (Szeged)194811229233263110030.17902 BiedlTLubiwAPetrickMSpriggsMMorphing orthogonal planar graph drawingsACM Trans. Algorithms (TALG)2013942931197871301.05237 Angelini, P., Da Lozzo, G., Frati, F., Lubiw, A., Patrignani, M., Roselli, V.: Optimal morphs of convex drawings. In: Symposium on Computational Geometry, volume 34 of LIPIcs, pp. 126–140. Schloss Dagstuhl-Leibniz-Zentrum fuer Informatik (2015) HongSNagamochiHConvex drawings of hierarchical planar graphs and clustered planar graphsJ. Discrete Algorithms201083282295265292010.1016/j.jda.2009.05.003 Alamdari, S., Angelini, P., Chan, T.M., Di Battista, G., Frati, F., Lubiw, A., Patrignani, M., Roselli, V., Singla, S., Wilkinson, B.T.: Morphing planar graph drawings with a polynomial number of steps. In: Khanna, S. (ed.) 24th Annual ACM-SIAM Symposium on Discrete Algorithms (SODA ’13), pp. 1656–1667 (2013) FratiFGudmundssonJWelzlEOn the number of upward planar orientations of maximal planar graphsTheor. Comput. Sci.20145443259322731810.1016/j.tcs.2014.04.024 L Kleist (714_CR26) 2019; 84 C Thomassen (714_CR34) 1983; 34 I Fáry (714_CR21) 1948; 11 G Di Battista (714_CR20) 1992; 7 P Angelini (714_CR4) 2014 G Di Battista (714_CR19) 1988; 61 H Tietze (714_CR35) 1914; 38 RF Cohen (714_CR15) 1995; 24 714_CR16 714_CR17 R Tamassia (714_CR33) 1986; 1 714_CR8 714_CR30 714_CR32 714_CR5 714_CR7 714_CR13 714_CR6 714_CR36 714_CR1 S Alamdari (714_CR2) 2017; 46 714_CR3 G Di Battista (714_CR18) 1999 P Bertolazzi (714_CR9) 1994; 4 A Garg (714_CR23) 2002; 31 SS Cairns (714_CR14) 1944; 51 J Hopcroft (714_CR25) 1974; 21 T Biedl (714_CR12) 2013; 9 P Bertolazzi (714_CR10) 1994; 12 714_CR28 K Mehlhorn (714_CR27) 1984 HL Smith (714_CR31) 1917; 19 P Bertolazzi (714_CR11) 1998; 27 S Hong (714_CR24) 2010; 8 F Frati (714_CR22) 2014; 544 P Rosenstiehl (714_CR29) 1986; 1 |
| References_xml | – reference: Tamassia, R., Tollis, I.G.: Algorithms for visibility representations of planar graphs. In: Monien, B., Vidal-Naquet, G. (eds.) STACS 86, 3rd Annual Symposium on Theoretical Aspects of Computer Science, Orsay, France, 16–18 January 1986, Proceedings, volume 210 of Lecture Notes in Computer Science, pp. 130–141. Springer (1986) – reference: Da Lozzo, G., Di Battista, G., Frati, F.: Extending upward planar graph drawings. In: WADS, volume 11646 of Lecture Notes in Computer Science, pp. 339–352. Springer (2019) – reference: Di BattistaGEadesPTamassiaRTollisIGGraph Drawing: Algorithms for the Visualization of Graphs1999Upper Saddle RiverPrentice-Hall1057.68653 – reference: GargATamassiaROn the computational complexity of upward and rectilinear planarity testingSIAM J. Comput.2002312601625186129210.1137/S0097539794277123 – reference: ThomassenCDeformations of plane graphsJ. Combin. Theory Ser. B198334324425771444810.1016/0095-8956(83)90038-2 – reference: Angelini, P., Frati, F., Patrignani, M., Roselli, V.: Morphing planar graph drawings efficiently. In: Wismath, S., Wolff, A. (eds.) Proceedings of 21st International Symposium on Graph Drawing (GD ’13), Lecture Notes in Computer Science. Springer (2013) – reference: BertolazziPDi BattistaGManninoCTamassiaROptimal upward planarity testing of single-source digraphsSIAM J. Comput.1998271132169161482110.1137/S0097539794279626 – reference: FáryIOn straight-line representation of planar graphsActa Scientiarum Mathematicarum (Szeged)194811229233263110030.17902 – reference: Alamdari, S., Angelini, P., Chan, T.M., Di Battista, G., Frati, F., Lubiw, A., Patrignani, M., Roselli, V., Singla, S., Wilkinson, B.T.: Morphing planar graph drawings with a polynomial number of steps. In: Khanna, S. (ed.) 24th Annual ACM-SIAM Symposium on Discrete Algorithms (SODA ’13), pp. 1656–1667 (2013) – reference: Biedl, T.C., Lubiw, A., Spriggs, M.J.: Morphing planar graphs while preserving edge directions. In: Healy, P., Nikolov, N.S. (eds.) 13th International Symposium on Graph Drawing (GD ’05), volume 3843 of Lecture Notes in Computer Science, pp. 13–24. Springer (2006) – reference: Angelini, P., Da Lozzo, G., Frati, F., Lubiw, A., Patrignani, M., Roselli, V.: Optimal morphs of convex drawings. In: Symposium on Computational Geometry, volume 34 of LIPIcs, pp. 126–140. Schloss Dagstuhl-Leibniz-Zentrum fuer Informatik (2015) – reference: Barrera-Cruz, F., Haxell, P., Lubiw, A.: Morphing planar graph drawings with unidirectional moves. In: Mexican Conference on Discrete Mathematics and Computational Geometry, pp. 57–65 (2013). arXiv:1411.6185 – reference: BiedlTLubiwAPetrickMSpriggsMMorphing orthogonal planar graph drawingsACM Trans. Algorithms (TALG)2013942931197871301.05237 – reference: SmithHLOn continuous representations of a square upon itselfAnn. Math.1917192137141150252210.2307/1967770 – reference: AlamdariSAngeliniPBarrera-CruzFChanTMDa LozzoGDi BattistaGFratiFHaxellPLubiwAPatrignaniMRoselliVSinglaSWilkinsonBTHow to morph planar graph drawingsSIAM J. Comput.2017462824852364063210.1137/16M1069171 – reference: HopcroftJTarjanREfficient planarity testingJ. ACM197421454956835938710.1145/321850.321852 – reference: TietzeHÜber stetige abbildungen einer quadratfläche auf sich selbstRendiconti del Circolo Matematico di Palermo191438124730410.1007/BF03015194 – reference: KleistLKlemzBLubiwASchlipfLStaalsFStrashDConvexity-increasing morphs of planar graphsComput. Geom.2019846988399770510.1016/j.comgeo.2019.07.007 – reference: Aichholzer, O., Aloupis, G., Demaine, E.D., Demaine, M.L., Dujmovic, V., Hurtado, F., Lubiw, A., Rote, G., Schulz, A., Souvaine, D.L., Winslow, A.: Convexifying polygons without losing visibilities. In: 23rd Canadian Conference on Computational Geometry (CCCG ’11) (2011) – reference: Di BattistaGTamassiaRTollisIGArea requirement and symmetry display of planar upward drawingsDiscrete Comput. Geom.199274381401114895310.1007/BF02187850 – reference: CairnsSSDeformations of plane rectilinear complexesAm. Math. Mon.19445152472521027310.1080/00029890.1944.11999082 – reference: MehlhornKData Structures and Algorithms: Multi-dimensional Searching and Computational Geometry1984BerlinSpringer0556.68003 – reference: Da Lozzo, G., Di Battista, G., Frati, F., Patrignani, M., Roselli, V.: Upward planar morphs. In: Biedl, T., Kerren, A. 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| Snippet | We prove that, given two topologically-equivalent upward planar straight-line drawings of an
n
-vertex directed graph
G
, there always exists a morph between... We prove that, given two topologically-equivalent upward planar straight-line drawings of an n-vertex directed graph G, there always exists a morph between... |
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| SubjectTerms | Algorithm Analysis and Problem Complexity Algorithms Computer Science Computer Systems Organization and Communication Networks Data Structures and Information Theory Equivalence Graph theory Mathematics of Computing Morphing Theory of Computation |
| Title | Upward Planar Morphs |
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