Drawing graphs using a small number of obstacles

Martin Balko, Josef Cibulka, Pavel Valtr

Research output: Chapter in Book/Report/Conference proceedingConference contributionpeer-review

4 Scopus citations

Abstract

An obstacle representation of a graph G is a set of points in the plane representing the vertices of G, together with a set of polygonal obstacles such that two vertices of G are connected by an edge in G if and only if the line segment between the corresponding points avoids all the obstacles. The obstacle number obs(G) of G is the minimum number of obstacles in an obstacle representation of G.We provide the first non-trivial general upper bound on the obsta- cle number of graphs by showing that every nvertex graph G satisfies obs(G) ≤ 2n log n. This refutes a conjecture of Mukkamala, Pach, and Pálvölgyi. For bipartite n-vertex graphs, we improve this bound to n-1. Both bounds apply even when the obstacles are required to be convex. We also prove a lower bound 2Ω(hn) on the number of n-vertex graphs with obstacle number at most h for h < n and an asymptotically matching lower bound Ω(n4/3M 2/3) for the complexity of a collection of M ≥ Ω(n) faces in an arrangement of n2 line segments with 2n endpoints.

Original languageEnglish
Title of host publicationGraph Drawing and Network Visualization - 23rd International Symposium, GD 2015, Revised Selected Papers
EditorsEmilio Di Giacomo, Anna Lubiw
PublisherSpringer Verlag
Pages360-372
Number of pages13
ISBN (Print)9783319272603
DOIs
StatePublished - 1 Jan 2015
Externally publishedYes
Event23rd International Symposium on Graph Drawing and Network Visualization, GD 2015 - Los Angeles, United States
Duration: 24 Sep 201526 Sep 2015

Publication series

NameLecture Notes in Computer Science (including subseries Lecture Notes in Artificial Intelligence and Lecture Notes in Bioinformatics)
Volume9411
ISSN (Print)0302-9743
ISSN (Electronic)1611-3349

Conference

Conference23rd International Symposium on Graph Drawing and Network Visualization, GD 2015
Country/TerritoryUnited States
CityLos Angeles
Period24/09/1526/09/15

Keywords

  • Arrangement of line segments
  • Geometric drawing
  • Obstacle number
  • Obstacle representation

ASJC Scopus subject areas

  • Theoretical Computer Science
  • Computer Science (all)

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