Morphing planar graphs while preserving edge directions

Abstract

Two straight-line drawings P, Q of a graph (V, E) are called parallel if, for every edge (u,v) ∈ E, the vector from u to v has the same direction in both P and Q. We study problems of the form: given simple, parallel drawings P, Q does there exist a continuous transformation between them such that intermediate drawings of the transformation remain simple and parallel with P (and Q)? We prove that a transformation can always be found in the case of orthogonal drawings; however, when edges are allowed to be in one of three or more slopes the problem becomes NP-hard. © Springer-Verlag Berlin Heidelberg 2005.