Graphs and Combinatorics

, Volume 29, Issue 5, pp 1207–1219 | Cite as

The Edge Rotation Graph

  • Javier Cano
  • José-Miguel Díaz-Báñez
  • Clemens Huemer
  • Jorge Urrutia
Original Paper

Abstract

Let P be a set of n points on the plane in general position, n ≥  3. The edge rotation graph \({\mathcal{E} \mathcal{R} \mathcal{G}(P,k)}\) of P is the graph whose vertices are the plane geometric graphs on P with exactly k edges, two of which are adjacent if one can be obtained from the other by an edge rotation. In this paper we study some structural properties of \({\mathcal{E} \mathcal{R} \mathcal{G}(P,k)}\) , such as its connectivity and diameter. We show that if the vertices of \({\mathcal{E} \mathcal{R} \mathcal{G}(P,k)}\) are not triangulations of P, then it is connected and has diameter O(n2). We also show that the chromatic number of \({\mathcal{E} \mathcal{R} \mathcal{G}(P,k)}\) is O(n), and show how to compute an implicit coloring of its vertices. We also study edge rotations in edge-labelled geometric graphs.

Keywords

Geometric graph Rotation Diameter 

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Copyright information

© Springer 2012

Authors and Affiliations

  • Javier Cano
    • 1
  • José-Miguel Díaz-Báñez
    • 2
  • Clemens Huemer
    • 3
  • Jorge Urrutia
    • 4
  1. 1.Posgrado en Ciencia e Ingeniería de la ComputaciónUniversidad Nacional Autónoma de MéxicoMexicoMexico
  2. 2.Departamento de Matemática Aplicada IIUniversidad de SevillaSevilleSpain
  3. 3.Departament de Matemàtica Aplicada IVUniversitat Politècnica de CatalunyaBarcelonaSpain
  4. 4.Instituto de MatemáticasUniversidad Nacional Autónoma de MéxicoMexicoMexico

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