Point-Set Embedding of Trees with Edge Constraints
Di Giacomo, Emilio and Didimo, Walter and Liotta, Giuseppe and Meijer, Henk and Wismath, Stephen (2008) Point-Set Embedding of Trees with Edge Constraints. In: Graph Drawing 15th International Symposium, GD 2007, September 24-26, 2007, Sydney, Australia , pp. 113-124 (Official URL: http://dx.doi.org/10.1007/978-3-540-77537-9_14).
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Given a graph G with n vertices and a set S of n points in the plane, a point-set embedding$ of G on S is a planar drawing such that each vertex of G is mapped to a distinct point of S. A geometric point-set embedding is a point-set embedding with no edge bends. This paper studies the following problem: The input is a set S of n points, a planar graph G with n vertices, and a geometric point-set embedding of a subgraph G' subset G on a subset of S. The desired output is a point-set embedding of G on S that includes the given partial drawing of G'. We concentrate on trees and show how to compute the output in O(n^2 log n) time and with at most 1 + 2 lceil k/2 rceil bends per edge, where k is the number of vertices of the given subdrawing. We also prove that there are instances of the problem which require at least k-3 bends for some of the edges.
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