Edge-Minimum Saturated k-Planar Drawings

Steven Chaplick, Fabian Klute, Irene Parada, Jonathan Rollin, Torsten Ueckerdt

Research output: Chapter in Book/Report/Conference proceedingChapterAcademic

Abstract

For a class \(\mathcal {d}\) of drawings of loopless (multi-)graphs in the plane, a drawing \(d \in \mathcal {d}\) is saturated when the addition of any edge to d results in \(d' \notin \mathcal {d}\)—this is analogous to saturated graphs in a graph class as introduced by turán (1941) and erdős, hajnal, and moon (1964). We focus on k-planar drawings, that is, graphs drawn in the plane where each edge is crossed at most k times, and the classes \(\mathcal {d}\) of all k-planar drawings obeying a number of restrictions, such as having no crossing incident edges, no pair of edges crossing more than once, or no edge crossing itself.while saturated k-planar drawings are the focus of several prior works, tight bounds on how sparse these can be are not well understood. We establish a generic framework to determine the minimum number of edges among all n-vertex saturated k-planar drawings in many natural classes. For example, when incident crossings, multicrossings and selfcrossings are all allowed, the sparsest n-vertex saturated k-planar drawings have \(\frac{2}{k - (k \bmod 2)} (n-1)\) edges for any \(k \ge 4\), while if all that is forbidden, the sparsest such drawings have \(\frac{2(k+1)}{k(k-1)}(n-1)\) edges for any \(k \ge 6\).
Original languageEnglish
Title of host publicationGraph Drawing and Network Visualization
Subtitle of host publication29th International Symposium, GD 2021, Tübingen, Germany, September 14–17, 2021, Revised Selected Papers
EditorsIgnaz Rutter, Helen C. Purchase
PublisherSpringer, Cham
Pages3-17
ISBN (Print)978-3-030-92930-5
DOIs
Publication statusPublished - 23 Dec 2021
EventThe 29th International Symposium on Graph Drawing and Network Visualization - Tübingen, Germany
Duration: 15 Sept 202117 Sept 2021
https://algo.inf.uni-tuebingen.de/gd2021//

Publication series

SeriesLecture Notes in Computer Science
Volume12868
ISSN0302-9743

Symposium

SymposiumThe 29th International Symposium on Graph Drawing and Network Visualization
Abbreviated titleGD 2021
Country/TerritoryGermany
CityTübingen
Period15/09/2117/09/21
Internet address

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