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Upward Planar Morphs

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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 \(\varOmega (n)\) morphing steps might be necessary for an upward planar morph between two topologically-equivalent upward planar straight-line drawings of an n-vertex path.

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Notes

  1. Two upward planar drawings \(\varGamma _0\) and \(\varGamma _1\) of a connected directed graph G are topologically-equivalent if, for each vertex v of G, the left-to-right order of the edges incoming into (outgoing from) v is the same in \(\varGamma _0\) as in \(\varGamma _1\).

  2. This insertion problem has been studied and solved in [2] for planar morphs of undirected graphs. Here we cannot immediately reuse the results in [2], as we need to preserve the upwardness of the drawing throughout the morph. However, the property that every drawing of \(G'\) in \({\mathcal{M}}'\) is upward significantly simplifies the problem of inserting v in \({\mathcal{M}}'\) so to obtain an upward planar morph of G.

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Acknowledgements

This research was supported in part by MIUR Project “MODE” under PRIN 20157EFM5C, by MIUR Project “AHeAD” under PRIN 20174LF3T8, by MIUR-DAAD JMP No. 34120, by H2020-MSCA-RISE project 734922—“CONNECT”, and by Roma Tre University Azione 4 Project “GeoView”.

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  1. Roma Tre University, Rome, Italy

    Giordano Da Lozzo, Giuseppe Di Battista, Fabrizio Frati, Maurizio Patrignani & Vincenzo Roselli

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  1. Giordano Da Lozzo
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  2. Giuseppe Di Battista
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Correspondence to Fabrizio Frati.

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A preliminary version of this paper appeared in [17].

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Da Lozzo, G., Di Battista, G., Frati, F. et al. Upward Planar Morphs. Algorithmica 82, 2985–3017 (2020). https://doi.org/10.1007/s00453-020-00714-6

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