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Spine Crossing Minimization in Upward Topological Book Embeddings Tamara Mchedlidze, Antonios Symvonis Department of Mathematics, National Technical University of Athens, Athens, Greece. 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 HPCCM problem 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-HPCCM problem. Acyclic-HPCCM Problem 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. Abstract Let G=(V,E) be an n node st-digraph. G has a crossing-optimal HP-completion set E c with Hamiltonian path P=(s = v 1, v 2,…, v n = t) such that, the corresponding optimal drawing Γ(G’) of G’=(V,E U E c ) 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 = v 1, v 2, …, v n = t). Theorem 1 An upward planar st- digraph G that is not Hamiltonian 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. 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 2 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. Theorem 3 Defenition: An st-polygon is a triangulated outerplanar st-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= (V ℓ U V r U {s,t}, E), where V ℓ and V r are 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 V ℓ are visited before the vertices of V r or, 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. 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 Two-Sided st-polygon and The resulting Hamiltonian path visits all vertices of V r and after all vertices of V ℓ Graph G c obtained by splitting the crossing edges. An upward topological book embedding of G c with its vertices placed on the spine in the order they appear on a hamiltonian path of G c.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. Optimal upward topological book embedding of G created from the drawing in the previous figure by deleting c 1, c 2, c 3, c 4 and merging the split edges of G. 1.Study of the Acyclic-HPCCM on the larger class of st-digraphs 2.Relaxing the requirement for G to be triangulated 3.Deriving HP-completion sets for the (Acyclic-) HPCCM problem that can have any number of crossing with the edges of graph G. Open problems and Future work 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.http://arxiv.org/abs/0807.2330 The resulting Hamiltonian path visits all vertices of V ℓ and after all vertices of V r One-Sided st-polygon

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