Stop Minding Your p's and q's: A Simplified O(n) Planar Embedding Algorithm

A graph is planar if it can be drawn on the plane with no crossing edges. There are several linear time planar embedding algorithms but all are considered by many to be quite complicated. This paper presents a new method for performing linear time planar graph embedding which avoids some of the complexities of previous approaches (including the need to rst st-number the vertices). Our new algorithm easily permits the extraction of a planar obstruction (a subgraph homeomorphic to K3;3 or K5) in O(n) time if the graph is not planar. Our algorithm is similar to the algorithm of Booth and Lueker which uses a data structure called a PQ-tree. The P-nodes in a PQ-tree represent parts of the partially embedded graph that can be permuted, and the Q-nodes represent parts that can be ipped. We avoid the use of P-nodes by not connecting pieces together until they become biconnected. We avoid Q nodes by using a data structure which allows biconnected components to be ipped in O(1) time. 1 Introduction An undirected graph G contains a set V of vertices and a set E of edges each which corresponds to an unordered pair of vertices from V. Throughout this paper, n is used to denote the number of vertices of a graph. Because loops (edges of the form (u; u)), and parallel edges (multiple edges with the same endpoints) provide no extra challenge, we assume that the graphs considered do not have loops or parallel edges (they are simple graphs). A graph is often drawn using points for the vertices and lines (possibly curved) for the edges. A geometric planar embedding of a graph is a drawing of the graph on a plane such that the vertices are placed in distinct positions and no two edges intersect except at common endpoints. Given a graph G, a planarity test algorithm determines if G has a planar embedding. A planar embedding algorithm also indicates the clockwise order of the neighbors of each vertex in such an embedding. To then generate a geometric planar embedding, one must choose vertex positions and edge shapes. This Supported by NSERC. is viewed as a separate problem, in part because it is application dependent. For example, our notion of what constitutes a suitable rendering of a graph may diier substantially if the graph represents an electronic circuit versus a hypertext book.