On ρ-Constrained Upward Topological Book Embeddings
نویسندگان
چکیده
Giordano, Liotta and Whitesides [1] developed an algorithm that, given an embedded planar st-digraph and a topological numbering ρ of its vertices, computes in O(n) time a ρ-constrained upward topological book embedding with at most 2n−4 spine crossings per edge. The number of spine crossings per edge is asymptotically worst case optimal. In this poster, we present improved results with respect to the number of spine crossings per edge and the time required to compute the book embedding. Firstly, for any embedded planar st-digraph G and any topological numbering ρ of its vertices, there exists a ρ-constrained upward topological book embedding with at most n− 3 spine crossings per edge and, moreover, n− 3 spine crossing per edge are required for some graphs. In this result, we allow edge (s, t) to be internal in the embedding of the graph. If edge (s, t) is always on the external face, the corresponding number of spine crossings reduces to at most n−4 and is worst case optimal. Secondly, a ρ-contrained upward topological book embedding with minimum number of spine crossings and at most n− 3 spine crossings per edge can be computed by an output sensitive algorithm in O(α+n) time, where α is the total number of spine crossings.
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