Embedding problems for paths with direction constrained edges

  • Authors:

  • Giuseppe Di Battista;Giuseppe Liotta;Anna Lubiw;Sue Whitesides

  • Affiliations:

  • Dipartimento di Informatica ed Automazione, Università di Roma Tre, Roma, Italy;Dipartimento di Ingegneria Elettronica e dell'Informazione, Università di Perugia, Perugia, Italy;Department of Computer Science, University of Waterloo, Waterloo, Canada;School of Computer Science, McGill University, Montreal, Que., Canada

  • Venue:
  • Theoretical Computer Science
  • Year:
  • 2002

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Abstract

We determine the reachability properties of the embeddings in R3 of a directed path, in the graph-theoretic sense, whose edges have each been assigned a desired direction (East, West, North, South, Up, or Down) but no length. We ask which points of R3 can be reached by the terminus of an embedding of such a path, by choosing appropriate positive lengths for the edges, if the embedded path starts at the origin, does not intersect itself, and respects the directions pre-assigned to its edges. This problem arises in the context of extending planar graph embedding techniques and VLSI rectilinear layout techniques from 2D to 3D. We give a combinatorial characterization of reachability that yields linear time recognition and layout algorithms. Finally, we extend our characterization to Rd, d 3.