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Nonplanar Graph Drawings with k Vertices per Face

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Graph-Theoretic Concepts in Computer Science (WG 2023)

Part of the book series: Lecture Notes in Computer Science ((LNCS,volume 14093))

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Abstract

The study of nonplanar graph drawings with forbidden or desired crossing configurations has a long tradition in geometric graph theory, and received an increasing attention in the last two decades, under the name of beyond-planar graph drawing. In this context, we introduce a new hierarchy of graph families, called \(k^+\)-real face graphs. For any integer \(k \ge 1\), a graph G is a \(k^+\)-real face graph if it admits a drawing \(\varGamma \) in the plane such that the boundary of each face (formed by vertices, crossings, and edges) contains at least k vertices of G. We give tight upper bounds on the maximum number of edges of \(k^+\)-real face graphs. In particular, we show that \(1^+\)-real face and \(2^+\)-real face graphs with n vertices have at most \(5n-10\) and \(4n-8\) edges, respectively. Also, if all vertices are constrained to be on the boundary of the external face, then \(1^+\)-real face and \(2^+\)-real face graphs have at most \(3n-6\) and \(2.5n-4\) edges, respectively. We also study relationships between \(k^+\)-real face graphs and beyond-planar graph families with hereditary property.

Research started at the Summer Workshop on Graph Drawing (SWGD) 2022, and partially supported by: (i) MIUR, grant 20174LF3T8 “AHeAD: efficient Algorithms for HArnessing networked Data”; (ii) Dipartimento di Ingegneria - Università degli Studi di Perugia, Ricerca di Base, grants RICBA21LG and RICBA22CB.

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Acknowledgments

We thank Vida Dujmović for valuable discussion.

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Authors and Affiliations

  1. Università degli Studi di Perugia, Perugia, Italy

    Carla Binucci, Walter Didimo, Giuseppe Liotta & Alessandra Tappini

  2. Università degli Studi Roma Tre, Rome, Italy

    Giuseppe Di Battista

  3. University of Sydney, Camperdown, Australia

    Seok-Hee Hong

  4. University of Tübingen, Tübingen, Germany

    Michael Kaufmann

  5. Carleton University, Ottawa, Canada

    Pat Morin

Authors
  1. Carla Binucci
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  2. Giuseppe Di Battista
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  3. Walter Didimo
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  4. Seok-Hee Hong
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  6. Giuseppe Liotta
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  7. Pat Morin
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  8. Alessandra Tappini
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Correspondence to Carla Binucci .

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Editors and Affiliations

  1. Durham University, Durham, UK

    Daniël Paulusma

  2. University of Fribourg, Fribourg, Fribourg, Switzerland

    Bernard Ries

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Binucci, C. et al. (2023). Nonplanar Graph Drawings with k Vertices per Face. In: Paulusma, D., Ries, B. (eds) Graph-Theoretic Concepts in Computer Science. WG 2023. Lecture Notes in Computer Science, vol 14093. Springer, Cham. https://doi.org/10.1007/978-3-031-43380-1_7

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  • DOI: https://doi.org/10.1007/978-3-031-43380-1_7

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