On the Relationship between $k$-Planar and $k$-Quasi Planar Graphs

Patrizio Angelini; Michael A. Bekos; Franz J. Brandenburg; Giordano Da Lozzo; Giuseppe Di Battista; Walter Didimo; Giuseppe Liotta; Fabrizio Montecchiani; Ignaz Rutter

arXiv · 1702.08716

On the Relationship between $k$-Planar and $k$-Quasi Planar Graphs

Patrizio Angelini·Michael A. Bekos·Franz J. Brandenburg·Giordano Da Lozzo·Giuseppe Di Battista·Walter Didimo·Giuseppe Liotta·Fabrizio Montecchiani·Ignaz Rutter

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Abstract

A graph is $k$-planar $(k \geq 1)$ if it can be drawn in the plane such that no edge is crossed more than $k$ times. A graph is $k$-quasi planar $(k \geq 2)$ if it can be drawn in the plane with no $k$ pairwise crossing edges. The families of $k$-planar and $k$-quasi planar graphs have been widely studied in the literature, and several bounds have been proven on their edge density. Nonetheless, only trivial results are known about the relationship between these two graph families. In this paper we prove that, for $k \geq 3$, every $k$-planar graph is $(k+1)$-quasi planar.

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Patrizio Angelini, Michael A. Bekos, Franz J. Brandenburg, Giordano Da Lozzo, Giuseppe Di Battista, Walter Didimo, Giuseppe Liotta, Fabrizio Montecchiani, Ignaz Rutter. 2017-02-28. On the Relationship between $k$-Planar and $k$-Quasi Planar Graphs. https://arxiv.org/abs/1702.08716

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