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Edge Partitions and Visibility Representations of 1-planar Graphs

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Beyond Planar Graphs

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Abstract

This chapter discusses the relationship between edge partitions and visibility representations of 1-planar graphs. Partitioning the edge set of a graph such that each partition set induces a simpler subgraph is a fundamental problem in graph theory, with applications in graph algorithms and graph drawing. For example, it is known that the edge set of every planar graph can be partitioned into two outerplanar graphs. A visibility representation of a graph is a classic drawing paradigm; it maps the vertices of the graph to geometric objects and the edges of the graph to lines of sight between pairs of objects. A classic result shows that every planar graph can be represented as a visibility representation such that the vertices are horizontal bars and the edges are vertical lines of sight between pairs of bars. While both edge partitions and visibility representations have been extensively studied for planar graphs, they recently attracted attention also for 1-planar graphs, i.e., those graphs that can be drawn in the plane such that each edge is crossed at most once. After giving an overview of 1-planarity, we survey the main results concerning edge partitions and visibility representations of 1-planar graphs, and we highlight an interesting interplay between them. In particular, we show how an edge partition of a 1-planar graph G into two planar subgraphs such that one of them has small vertex degree can be used to construct a visibility representation of G in which vertices are orthogonal polygons with few reflex corners each. Finally, we conclude this chapter with a selection of open problems related to the covered topics.

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Authors and Affiliations

  1. Dipartimento di Ingegneria, Università degli Studi di Perugia, Via G. Duranti 93, 06125, Perugia, Italy

    Giuseppe Liotta & Fabrizio Montecchiani

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  1. Giuseppe Liotta
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  2. Fabrizio Montecchiani
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Correspondence to Fabrizio Montecchiani .

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  1. School of Computer Science, University of Sydney, Sydney, NSW, Australia

    Seok-Hee Hong

  2. School of Science and Technology, Kwansei Gakuin University, Sanda, Hyogo, Japan

    Takeshi Tokuyama

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Liotta, G., Montecchiani, F. (2020). Edge Partitions and Visibility Representations of 1-planar Graphs. In: Hong, SH., Tokuyama, T. (eds) Beyond Planar Graphs. Springer, Singapore. https://doi.org/10.1007/978-981-15-6533-5_6

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  • DOI: https://doi.org/10.1007/978-981-15-6533-5_6

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