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Min-$k$-planar Drawings of Graphs

Carla Binucci, Aaron Büngener, Giuseppe Di Battista, Walter Didimo, Vida Dujmović, Seok-Hee Hong, Michael Kaufmann, Giuseppe Liotta, Pat Morin, Alessandra Tappini

TL;DR

This paper introduces min-$k$-planar graphs, a relaxation of $k$-planarity that permits heavy edges while requiring that any crossing pair includes a light edge with at most $k$ crossings. It develops general edge-density bounds and tight results for $k\in\{1,2,3\}$, along with bounds on the number of heavy edges and detailed inclusion relations with related beyond-planar graph families. The authors employ a combination of crossing lemmas and discharging techniques to derive these bounds and to analyze the interplay between density and the presence of heavy edges. They also establish how min-$k$-planarity relates to $k$-gap-planar and $(k+2)$-quasiplanar classes, revealing both inclusions and incomparabilities with other well-studied beyond-planar families. Overall, the work advances the structural understanding and density limits of beyond-planar graphs and clarifies how the min-$k$-planar framework interacts with established graph-drawing hierarchies.

Abstract

The study of nonplanar drawings of graphs with restricted crossing configurations is a well-established topic in graph drawing, often referred to as beyond-planar graph drawing. One of the most studied types of drawings in this area are the $k$-planar drawings $(k \geq 1)$, where each edge cannot cross more than $k$ times. We generalize $k$-planar drawings, by introducing the new family of min-$k$-planar drawings. In a min-$k$-planar drawing edges can cross an arbitrary number of times, but for any two crossing edges, one of the two must have no more than $k$ crossings. We prove a general upper bound on the number of edges of min-$k$-planar drawings, a finer upper bound for $k=3$, and tight upper bounds for $k=1,2$. Also, we study the inclusion relations between min-$k$-planar graphs (i.e., graphs admitting min-$k$-planar drawings) and $k$-planar graphs. In our setting we only allow simple drawings, that is, any two edges cross at most once, no two adjacent edges cross, and no three edges intersect at a common crossing point.

Min-$k$-planar Drawings of Graphs

TL;DR

This paper introduces min--planar graphs, a relaxation of -planarity that permits heavy edges while requiring that any crossing pair includes a light edge with at most crossings. It develops general edge-density bounds and tight results for , along with bounds on the number of heavy edges and detailed inclusion relations with related beyond-planar graph families. The authors employ a combination of crossing lemmas and discharging techniques to derive these bounds and to analyze the interplay between density and the presence of heavy edges. They also establish how min--planarity relates to -gap-planar and -quasiplanar classes, revealing both inclusions and incomparabilities with other well-studied beyond-planar families. Overall, the work advances the structural understanding and density limits of beyond-planar graphs and clarifies how the min--planar framework interacts with established graph-drawing hierarchies.

Abstract

The study of nonplanar drawings of graphs with restricted crossing configurations is a well-established topic in graph drawing, often referred to as beyond-planar graph drawing. One of the most studied types of drawings in this area are the -planar drawings , where each edge cannot cross more than times. We generalize -planar drawings, by introducing the new family of min--planar drawings. In a min--planar drawing edges can cross an arbitrary number of times, but for any two crossing edges, one of the two must have no more than crossings. We prove a general upper bound on the number of edges of min--planar drawings, a finer upper bound for , and tight upper bounds for . Also, we study the inclusion relations between min--planar graphs (i.e., graphs admitting min--planar drawings) and -planar graphs. In our setting we only allow simple drawings, that is, any two edges cross at most once, no two adjacent edges cross, and no three edges intersect at a common crossing point.
Paper Structure (3 sections, 1 figure)

This paper contains 3 sections, 1 figure.

Table of Contents

  1. Introduction
  2. Basic Definitions
  3. Edge Density of Min-$k$-planar Graphs

Figures (1)

  • Figure 1: Two drawings of the same portion of a graph: (a) is 2-planar and has 10 crossings; (b) is min-2-planar, is not 2-planar, and has 12 crossings; it contains two "heavy" edges incident to vertex 6, each with several crossings.

Theorems & Definitions (1)

  • definition thmcounterdefinition