Acyclic-HPCCM Problem

1. Spine Crossing Minimization in Upward Topological Book EmbeddingsTamara Mchedlidze , AntoniosSymvonisDepartment of Mathematics, National Technical University of Athens, Athens, Greece. Theorem 1 Some ideas of ours linear-time algorithm that solves the Acyclic-HPCCM problem with at most one crossing per edge for outerplanar triangulated st-digraphs Abstract Let G=(V,E) be an n node st-digraph. G has a crossing-optimal HP-completion set Ecwith Hamiltonian path P=(s=v1, v2,…, vn=t) such that, the corresponding optimal drawing Γ(G’) of G’=(V,E UEc) has c crossings iff G has an optimal (with respect to the number of spine crossings) upward topological book embedding with c spine crossings where the vertices appear on the spine in the order Π=(s=v1, v2, … , vn=t). Given an embedded planar acyclic digraph, we define the problem of Acyclic Hamiltonian Path Completion with Crossing Minimization (Acyclic-HPCCM) and establish an equivalence between it and the problem of determining an upward topological book embedding with minimum number of spine crossings. By developing a linear-time algorithm that solves the Acyclic-HPCCM problem with at most one crossing per edge for outerplanar triangulated st-digraphs, we infer for this class of graphs an optimal (with respect to spine crossings) upward topological book embedding with at most one spine crossing per edge. Defenition: An st-polygon is a triangulated outerplanarst-digraph that always contains edge (s,t) connecting its source to its sink. There are two kinds of st-polygons: It’s easy to compute an optimal (with respect to the number of edge crossing) HP-completion set of an st-polygon: Lemma: Assume an st-polygon R= (VℓUVrU {s,t}, E), where Vℓand Vrare the vertices at its left and right border respectively. In a crossing-optimal acyclic HP-completion set of R with at most 1 edge crossing per initial edge, the vertices of Vℓ are visited before the vertices of Vror, vice versa. So, the two possible solutions are: Given an outerplanar triangulated st-digraph G, the st-polygon decomposition of G is defined to be the total order of its maximal st-polygons and remaining vertices. Based on the decomposition properties, we develop a dynamic programming linear-time algorithm, that solves the Acyclic-HPCCM problem with at most one crossing per edge of G. Two-Sided st-polygon One-Sided st-polygon and Acyclic-HPCCM Problem Theorem 2 • We define the Hamiltonian Path Completion with Crossing Minimization problem as follows: • Given an embedded planar graph G=(V,E), directed or undirected, one non-negative integer c, and two vertices s, t in V, the HPCCMproblem asks whether there exists: • An edge superset E’ containing E (the set E\E’ is called HP-completion set) and • A drawing Γ’ of graph G’= (V, E’) • such that: • G’ has a Hamiltonian path from vertex s to vertex t, • G’ has at most c edge crossings, and • G’ preserves the embedded planar graph G. • 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-HPCCMproblem. Given an n node outerplanar triangulated st-digraph G, a crossing-optimal HP-completion set for G with at most one crossing per edge can be computed in O(n) time. Theorem 3 Given an n node outerplanar triangulated st-digraph G, an upward topological book embedding for G with minimum number of spine crossings and at most one spine crossing per edge can be computed in O(n)time. The resulting Hamiltonian path visits all vertices of Vr and after all vertices of Vℓ The resulting Hamiltonian path visits all vertices of Vℓand after all vertices of Vr Optimal upward topological book embedding of G created from the drawing in the previous figure by deleting c1, c2, c3, c4 and merging the split edges of G. An upward topological book embedding of Gcwith its vertices placed on the spine in the order they appear on a hamiltonian path of Gc .The edges appearing on the left(resp. right) side of the Hamiltonian path (as traveling from s to t) are placed on the left(resp.right) half-plane. Graph G is augmented by the edges of an optimal HP-completion set (bold red edges) produced by our algorithm. The created Hamiltonian path is drawn with bold edges. Graph Gcobtained by splitting the crossing edges. An upward planar st-digraph G that is not Hamiltonian Open problems and Future work Study of the Acyclic-HPCCM on the larger class of st-digraphs Relaxing the requirement for G to be triangulated Deriving HP-completion sets for the (Acyclic-) HPCCM problem that can have any number of crossing with the edges of graph G. More information can be found in: “Optimal Acyclic Hamiltonian Path Completion for Outerplanar Triangulated st-Digraphs (with Application to Upward Topological Book Embeddings)” at http://arxiv.org/abs/0807.2330.