Point-set embedding of trees with edge constraints

Abstract

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(n2 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. © 2008 Springer-Verlag Berlin Heidelberg.