Special Issue on Selected Papers from the Third Annual Workshop on Algorithms and Computation (WALCOM 2009)
Crossing-Optimal Acyclic HP-Completion for Outerplanar st-Digraphs
Vol. 15, no. 3, pp. 373-415, 2011. Regular paper.
Abstract Given an embedded planar acyclic digraph G, we define the problem of acyclic hamiltonian path completion with crossing minimization (acyclic-HPCCM) to be the problem of determining a hamiltonian path completion set of edges such that, when these edges are embedded on G, they create the smallest possible number of edge crossings and turn G to a hamiltonian acyclic digraph. Our results include:
  1. We provide a characterization under which a planar st-digraph G is hamiltonian.
  2. For an outerplanar st-digraph G, we define the st-Polygon decomposition of G and, based on its properties, we develop a linear-time algorithm that solves the acyclic-HPCCM problem.
  3. For the class of planar st-digraphs, we establish an equivalence between the acyclic-HPCCM problem and the problem of determining an upward 2-page topological book embedding with minimum number of spine crossings. We obtain (based on this equivalence) for the class of outerplanar st-digraphs, an upward topological 2-page book embedding with minimum number of spine crossings.
To the best of our knowledge, it is the first time that edge-crossing minimization is studied in conjunction with the acyclic hamiltonian completion problem and the first time that an optimal algorithm with respect to spine crossing minimization is presented for upward topological book embeddings.
Submitted: May 2009.
Reviewed: January 2010.
Revised: April 2010.
Accepted: October 2010.
Final: November 2010.
Published: July 2011.
Communicated by Sandip Das and Ryuhei Uehara
article (PDF)