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Planar Bus Graphs

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Abstract

Bus graphs are used for the visualization of hypergraphs, for example in VLSI design. Formally, they are specified by bipartite graphs \(G=(B \cup V,E)\). The bus vertices B are realized by horizontal and vertical segments, and the connector vertices V are realized by points and connected orthogonally to the bus segments without any bend; this is called bus realization. The decision whether a bipartite graph admits a bus realization, where connections may cross, is NP-complete. In this paper we show that in contrast the question whether a planar bipartite graph admits a planar bus realization can be answered in polynomial time. First we deal with plane instances, i.e., with the case where a planar embedding is prescribed. We identify three necessary conditions on the partition \(B=B_{\mathrm {V}}\cup B_{\mathrm {H}}\) of the bus vertices, here \(B_{\mathrm {V}}\) denotes the vertical and \(B_{\mathrm {H}}\) the horizontal buses. We provide a test whether a good partition, i.e., a partition obeying these conditions, exists. The test is based on the computation of a maximum matching on some auxiliary graph. Given a good partition we can construct a non-crossing realization of the bus graph on an \(O(n)\times O(n)\) grid in linear time. In the second part we use SPQR-trees to solve the problem for general planar bipartite graphs.

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Notes

  1. A planar embedding is a drawing of the vertices together with a description of the cyclic order of their adjacent vertices. Often a drawing with curves representing the edges is used to illustrate an embedding.

  2. In [24] a slightly faster randomized algorithm for planar graphs has been proposed.

  3. Some descriptions also use type Q nodes, so that SPQR are the possible types.

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Acknowledgements

We thank the anonymous reviewers for their valuable comments and their persistence. With their help the quality of the paper has been improved significantly.

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Authors and Affiliations

  1. Wilhelm-Schickard-Institut für Informatik, Universität Tübingen, Tübingen, Germany

    Till Bruckdorfer & Michael Kaufmann

  2. Institut für Mathematik, Technische Universität Berlin, 10623, Berlin, Germany

    Stefan Felsner

Authors
  1. Till Bruckdorfer
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  2. Stefan Felsner
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  3. Michael Kaufmann
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Corresponding author

Correspondence to Michael Kaufmann.

Additional information

T. Bruckdorfer, M. Kaufmann: Partially supported by EuroGIGA project GraDR 10-EuroGIGA-OP-003 S. Felsner: Partially supported by DFG Grant FE-340/7-2 and ESF EuroGIGA project Compose.

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Bruckdorfer, T., Felsner, S. & Kaufmann, M. Planar Bus Graphs. Algorithmica 80, 2260–2285 (2018). https://doi.org/10.1007/s00453-017-0321-5

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