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Minimizing an Uncrossed Collection of Drawings

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Graph Drawing and Network Visualization (GD 2023)

Part of the book series: Lecture Notes in Computer Science ((LNCS,volume 14465))

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Abstract

In this paper, we introduce the following new concept in graph drawing. Our task is to find a small collection of drawings such that they all together satisfy some property that is useful for graph visualization. We propose investigating a property where each edge is not crossed in at least one drawing in the collection. We call such collection uncrossed. This property is motivated by a quintessential problem of the crossing number, where one asks for a drawing where the number of edge crossings is minimum. Indeed, if we are allowed to visualize only one drawing, then the one which minimizes the number of crossings is probably the neatest for the first orientation. However, a collection of drawings where each highlights a different aspect of a graph without any crossings could shed even more light on the graph’s structure.

We propose two definitions. First, the uncrossed number, minimizes the number of graph drawings in a collection, satisfying the uncrossed property. Second, the uncrossed crossing number, minimizes the total number of crossings in the collection that satisfy the uncrossed property. For both definitions, we establish initial results. We prove that the uncrossed crossing number is NP-hard, but there is an \(\textsf{FPT}\) algorithm parameterized by the solution size.

The full version of this paper is available on arXiv [27].

T. M. was supported by Polish National Science Centre SONATA-17 grant number 2021/43/D/ST6/03312.

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Notes

  1. 1.

    Proofs marked by *are available in the appendix of [27].

  2. 2.

    There are some possible exceptions in case of \(\theta (K_{m,n})\) where the exact value is not known; see [36] for further discussion.

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Authors and Affiliations

  1. Faculty of Informatics of Masaryk University, Brno, Czech Republic

    Petr Hliněný

  2. Institute of Informatics, Faculty of Mathematics, Informatics and Mechanics, University of Warsaw, Warsaw, Poland

    Tomáš Masařík

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  1. Petr Hliněný
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Correspondence to Tomáš Masařík .

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  1. University Ioannina, Ioannina, Greece

    Michael A. Bekos

  2. University of Ioannina, Osnabrück, Germany

    Markus Chimani

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Hliněný, P., Masařík, T. (2023). Minimizing an Uncrossed Collection of Drawings. In: Bekos, M.A., Chimani, M. (eds) Graph Drawing and Network Visualization. GD 2023. Lecture Notes in Computer Science, vol 14465. Springer, Cham. https://doi.org/10.1007/978-3-031-49272-3_8

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