Drawing Graphs as Spanners

Oswin Aichholzer, Manuel Borrazzo, Prosenjit Bose, Jean Cardinal, Fabrizio Frati*, Pat Morin, Birgit Vogtenhuber

*Korrespondierende/r Autor/-in für diese Arbeit

Publikation: Beitrag in einer FachzeitschriftArtikelBegutachtung

Abstract

We study the problem of embedding graphs in the plane as good geometric spanners. That is, for a graph G, the goal is to construct a straight-line drawing Γ of G in the plane such that, for any two vertices u and v of G, the ratio between the minimum length of any path from u to v and the Euclidean distance between u and v is small. The maximum such ratio, over all pairs of vertices of G, is the spanning ratio of Γ. First, we show that deciding whether a graph admits a straight-line drawing with spanning ratio 1, a proper straight-line drawing with spanning ratio 1, and a planar straight-line drawing with spanning ratio 1 are NP-complete, ∃ R-complete, and linear-time solvable problems, respectively, where a drawing is proper if no two vertices overlap and no edge overlaps a vertex. Second, we show that moving from spanning ratio 1 to spanning ratio 1 + ϵ allows us to draw every graph. Namely, we prove that, for every ϵ> 0 , every (planar) graph admits a proper (resp. planar) straight-line drawing with spanning ratio smaller than 1 + ϵ. Third, our drawings with spanning ratio smaller than 1 + ϵ have large edge-length ratio, that is, the ratio between the length of the longest edge and the length of the shortest edge is exponential. We show that this is sometimes unavoidable. More generally, we identify having bounded toughness as the criterion that distinguishes graphs that admit straight-line drawings with constant spanning ratio and polynomial edge-length ratio from graphs that require exponential edge-length ratio in any straight-line drawing with constant spanning ratio.

Originalspracheenglisch
Seiten (von - bis)774-795
Seitenumfang22
FachzeitschriftDiscrete and Computational Geometry
Jahrgang68
Ausgabenummer3
DOIs
PublikationsstatusVeröffentlicht - Okt. 2022

ASJC Scopus subject areas

  • Theoretische Informatik
  • Geometrie und Topologie
  • Diskrete Mathematik und Kombinatorik
  • Theoretische Informatik und Mathematik

Fields of Expertise

  • Information, Communication & Computing

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