No-Bend Orthogonal Drawings and No-Bend Orthogonally Convex Drawings of Planar Graphs (Extended Abstract)

Abstract

A plane graph is a planar graph with a fixed planar embedding in the plane. In an orthogonal drawing of a plane graph each vertex is drawn as a point and each edge is drawn as a sequence of vertical and horizontal line segments. A bend is a point at which the drawing of an edge changes its direction. A necessary and sufficient condition for a plane graph of maximum degree 3 to have a no-bend orthogonal drawing is known which leads to a linear-time algorithm to find such a drawing of a plane graph, if it exists. A planar graph G has a no-bend orthogonal drawing if any of the plane embeddings of G has a no-bend orthogonal drawing. Since a planar graph G of maximum degree 3 may have an exponential number of planar embeddings, determining whether G has a no-bend orthogonal drawing or not using the known algorithm for plane graphs takes exponential time. The best known algorithm takes O(n^2) time for finding a no-bend orthogonal drawing of a biconnected planar graph of maximum degree 3. In this paper we give a linear-time algorithm to determine whether a biconnected planar graph G of maximum degree 3 has a no-bend orthogonal drawing or not and to find such a drawing of G, if it exists. We also give a necessary and sufficient condition for a biconnected planar graph G of maximum degree 3 to have a no-bend “orthogonally convex” drawing D; where any horizontal and vertical line segment connecting two points in a facial polygon P in D lies totally within P. Our condition leads to a linear-time algorithm for finding such a drawing, if it exists.