Point-Set Embedding of Trees with Edge Constraints
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′ ⊂ 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 logn) time and with at most 1 + 2 ⌈k/2 ⌉ 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.
KeywordsPlanar Graph Convex Polygon Dual Graph Outerplanar Graph Planar Drawing
- 1.Badent, M., Di Giacomo, E., Liotta, G.: Drawing colored graphs on colored points. In: WADS 2007. LNCS, Springer, Heidelberg (2007)Google Scholar
- 17.Preparata, F.P., Shamos, M.I.: Computational Geometry: An Introduction, 3rd edn. Springer, Heidelberg (1990)Google Scholar