Constrained point-set embeddability of planar graphs

Abstract

This paper starts the investigation of a constrained version of the point-set embeddability problem. Let G∈=∈(V,E) be a planar graph with n vertices, G?∈=∈(V?E?) a subgraph of G, and S a set of n distinct points in the plane. We study the problem of computing a point-set embedding of G on S subject to the constraint that G? is drawn with straight-line edges. Different drawing algorithms are presented that guarantee small curve complexity of the resulting drawing, i.e. a small number of bends per edge. It is proved that: (i) If G? is an outerplanar graph and S is any set of points in convex position, a point-set embedding of G on S can be computed such that the edges of E∈? ∈E? have at most 4 bends each. (ii) If S is any set of points in general position and G? is a face of G or if it is a simple path, the curve complexity of the edges of E∈? ∈E? is at most 8. (iii) If S is in general position and G? is a set of k disjoint paths, the curve complexity of the edges of E∈? ∈E? is O(2 k ). © 2009 Springer Berlin Heidelberg.