Weak unit disk and interval representation of graphs

M. J. Alam, Stephen G Kobourov, S. Pupyrev, J. Toeniskoetter

Research output: Chapter in Book/Report/Conference proceedingConference contribution

1 Scopus citations

Abstract

We study a variant of intersection representations with unit balls: unit disks in the plane and unit intervals on the line. Given a planar graph and a bipartition of the edges of the graph into near and far edges, the goal is to represent the vertices of the graph by unit-size balls so that the balls for two adjacent vertices intersect if and only if the corresponding edge is near. We consider the problem in the plane and prove that it is NP-hard to decide whether such a representation exists for a given edgepartition. On the other hand, we show that series-parallel graphs (which include outerplanar graphs) admit such a representation with unit disks for any near/far bipartition of the edges. The unit-interval on the line variant is equivalent to threshold graph coloring, in which context it is known that there exist girth-3 planar graphs (even outerplanar graphs) that do not admit such coloring. We extend this result to girth-4 planar graphs. On the other hand, we show that all triangle-free outerplanar graphs and all planar graphs with maximum average degree less than 26/11 have such a coloring, via unit-interval intersection representation on the line. This gives a simple proof that all planar graphs with girth at least 13 have a unit-interval intersection representation on the line.

Original languageEnglish (US)
Title of host publicationGraph-Theoretic Concepts in Computer Science - 41st International Workshop, WG 2015, Revised Papers
PublisherSpringer Verlag
Pages237-251
Number of pages15
Volume9224
ISBN (Print)9783662531730
DOIs
StatePublished - 2016
Event41st International Workshop on Graph-Theoretic Concepts in Computer Science, WG 2015 - Garching, Germany
Duration: Jun 17 2015Jun 19 2015

Publication series

NameLecture Notes in Computer Science (including subseries Lecture Notes in Artificial Intelligence and Lecture Notes in Bioinformatics)
Volume9224
ISSN (Print)03029743
ISSN (Electronic)16113349

Other

Other41st International Workshop on Graph-Theoretic Concepts in Computer Science, WG 2015
CountryGermany
CityGarching
Period6/17/156/19/15

ASJC Scopus subject areas

  • Theoretical Computer Science
  • Computer Science(all)

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