Curve-constrained drawings of planar graphs

  • Authors:

  • Emilio Di Giacomo;Walter Didimo;Giuseppe Liotta;Stephen K. Wismath

  • Affiliations:

  • Dipartimento di Ingegneria Elettronica e dell'Informazione, Università di Perugia, Italy;Dipartimento di Ingegneria Elettronica e dell'Informazione, Università di Perugia, Italy;Dipartimento di Ingegneria Elettronica e dell'Informazione, Università di Perugia, Italy;Department of Mathematics and Computer Science, University of Lethbridge, Alberta, Canada

  • Venue:
  • Computational Geometry: Theory and Applications
  • Year:
  • 2005

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Abstract

Let C be the family of 2D curves described by concave functions, let G be a planar graph, and let L be a linear ordering of the vertices of G. L is a curve embedding of G if for any given curve Λ ∈ C there exists a planar drawing of G such that: (i) the vertices are constrained to be on Λ with the same ordering as in L, and (ii) the edges are polylines with at most one bend. Informally speaking, a curve embedding can be regarded as a two-page book embedding in which the spine is bent. Although deciding whether a graph has a two-page book embedding is an NP-hard problem, in this paper it is proven that every planar graph has a curve embedding which can be computed in linear time. Applications of the concept of curve embedding to upward drawability and point-set embeddability problems are also presented.