Crossing-Optimal Acyclic HP-Completion for Outerplanar st-Digraphs

Given an embedded planar acyclic digraph G, the acyclic hamiltonian path completion with crossing minimization (AcyclicHPCCM) problem is to determine 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. In this paper, we present a linear time algorithm which solves the Acyclic-HPCCM problem on any outerplanar st-digraph G. The algorithm is based on properties of the optimal solution and an st-polygon decomposition of G. As a consequence of our result, we can obtain for the class of outerplanar st-digraphs upward topological 2-page book embeddings with minimum number of spine crossings.