Embedding problems for paths with direction constrained edges

Abstract

We determine the reachability properties of the embeddings in R3 of a directed path, in the graph theoretic sense, whose edges have each been assigned a desired direction (East, West, North, South, Up, or Down) but no length. We ask which points of R3 can be reached by the terminus of an embedding of such a path, by choosing appropriate positive lengths for the edges, if the embedded path starts at the origin, does not intersect itself, and respects the directions assigned to its edges. This problem arises in the context of extending planar graph embedding techniques and VLSI rectilinear layout techniques from 2D to 3D. We give combinatorial characterizations of reachability that yield linear time recognition and layout algorithms.