Spine Crossing Minimization in Upward Topological Book Embeddings

An upward topological book embedding of a planar st-digraph G is an upward planar drawing of G such that its vertices are aligned along the vertical line, called the spine, and each edge is represented as a simple Jordan curve which is divided by the intersections with the spine (spine crossings) into segments such that any two consecutive segments are located at opposite sides of the spine. When we treat the problem of obtaining an upward topological book embedding as an optimization problem, we are naturally interested in embeddings with the minimum possible number of spine crossing. We define the problem of HP-completion with crossing minimization problem (for short, HPCCM ) as follows: Given an embedded planar graph G = (V,E), directed or undirected, one non-negative integer c, and two vertices s, t ∈ V , the HPCCM problem asks whether there exists a superset E′ containing E and a drawing Γ (G′) of graph G′ = (V,E′) such that (i) G′ has a hamiltonian path from vertex s to vertex t, (ii) Γ (G′) has at most c edge crossings, and (iii) Γ (G′) preserves the embedded planar graphG. When the input digraph G is acyclic, we can insist on HP-completion sets which leave the HP-completed digraph G′ also acyclic. We refer to this version of the problem as the Acyclic-HPCCM problem.