On the Cutting Edge: Simplified O(n) Planarity by Edge Addition

Journal of Computer and System Sciences, 1985

Moreover this embedding algorithm can find all the embeddings of a planar graph. 0 1985 Academic Press, hc. 1. INTRoDUCTIoN Planarity testing, that is, determining whether a given graph is planar or not, has many applications, such as the design of VLSI circuits and determining isomorphism of chemical structures. Two planarity testing algorithms bf different types are known, both running in linear' time. One is called a "path addition algorithm," and the other a "vertex addition algorithm." These terms "path addition" and "vertex addition" express well the principles of the algorithms. The path addition algorithm was first presented by Auslander and Parter Cl] and Goldstein [6], and improved later into a linear algorithm by Hopcroft and Tarjan [9]. The vertex addition algorithm was first presented by Lempel, Even, and Cederbaum [lo], and improved later into a linear algorithm by Booth and Lueker 54

Electronics and Communications in Japan (Part I: Communications), 1984

The problems of t e s t i n g the p l a n a r i t y of a graph and of embedding a planar graph i n a plane a r i s e i n many applications. This paper presents a simple l i n e a r algorithm f o r the l a t t e r problem. on the "vertex-addition algorithm" of Lempel, Even and Cederbaum f o r planarity t e s t i n g and i s a s l i g h t modification of Booth and Lueker's implementation of t h e t e s t i n g algorithm using a PQ-tree. bedding algorithm known a s t h e "path-addition" algorithm of Hopcroft and Tarjan, our algorithm i s conceptually simple and easy t o understand or implement.

Symposium on Discrete Algorithms, 1999

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 ah 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 first st-number the vertices). Our new algorithm easily permits the extraction of a planar obstruction (a subgraph homeomorphic to Ks,s or Ks) in O(n) time if the graph is not planar. Lueker which uses a data structure called a PQ-tree. The Pnodes in a P&-tree represent parts of the partially embedded graph that can be permuted, and the Q-nodes represent parts that can be flipped. 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 fiipped in O(1) time.

Journal of Computer and System Sciences, 1988

Planarity is a classical property of graphs, dating back to 1736, when Euler formulated the concept. A drawing of a graph on a plane in which no edges cross is called a planar embedding. A graph for which such an embedding exists is called a planar graph. The search for an efficient algorithm to decide planarity and find a planar embedding culminated in Hopcroft and Tarjan's linear-time algorithm ([101). Continuing in this tradition, we have developed a very efficient parallel algorithm for this problem. Our new algorithm finds a planar embedding for an n-node graph (or reports that none exists) in O(log 2 n) time using only n processors of an exclusive-write, concurrent-read PRAM. Thus it achieves near-optimal speedup. The fact that only n processors are needed for our algorithm makes it feasible to implement on a real parallel computer. In contrast, the previous best parallel algorithm for testing planarity, due to Ja'Ja' and Simon ([12]), reduced the problem to solving linear systems, and hence required M(n) processors, where M(n) is the number of operations required to multiply two n x n matrices. Ja'Ja' and Simon's algorithm was important because it showed that planarity could be decided quickly in parallel. However, such a large processor bound makes their algorithm infeasible. For any n large enough that a parallel algorithm would be preferred to the linear-time sequential algorithm (e.g., n = 5, 000), M(n) far exceeds the number of processors on any realistic parallel computer (e.g., M(5,000) > 125,000,000,000). Our planarity algorithm, on the other hand, would achieve the same time bound with only 5,000 processors. In general, once a problem has been shown to be in NC, it is important for designers of * finding a depth-first search tree of a planar graph in O(log 3 n) time using n processors (using ideas due to Justin Smith, combined with techniques of [23]). * finding an outerplanar embedding of an outerplanar graph in O(log 2 n) time using n processors, and finding an 0(1) separator of an outerplanar graph. Because our algorithm does not rule out any embeddings, it can be used to enumerate all possible planar embeddings of a graph at a rate of O(log 2 n) time per embedding, using n processors. 1.2 PQ-Trees Our parallel planarity algorithm is rare among parallel algorithms in that it uses a sophisticated data structure. We have parallelized the PQ-tree data structure, due to Booth and Lueker ([4]), giving efficient parallel algorithms for manipulating PQ-trens. No parallel algorithms for PQ-trees existed previously. We define three operations on PQ-trees, multiple-disjoint-reduction, join, and intersection, and give linear-processor parallel algorithms for these operations. We use PQ-trees for representing sets of graph embeddings. However, PQ-trees are generally useful for representing large sets of orderings subject to adjacency constraints. Booth and Lueker use PQ-trees in efficient sequential algorithms for recognizing (0,1)-matrices with the consecutive one's property, and in recognizing and testing isomorphism of interval graphs. Using our parallel algorithms for PQ-trees, one

IEEE Transactions on Computer-Aided Design of Integrated Circuits and Systems, 1989

Abstmct-In this paper we present two O ( n * ) planarization algorithms-PLANARIZE and MAXIMAL-PLANARIZE. These algorithms are based on Lempel, Even, and Cederbaum's planarity testing algorithm [9] and its implementation using PQ-trees [8]. Algorithm PLANARIZE is for the construction of a spanning planar subgraph of an n-vertex nonplanar graph. This algorithm proceeds by embedding one vertex at a time and, at each step, adds the maximum number of edges possible without creating nonplanarity of the resultant graph. Given a biconnected spanning planar subgraph G,, of a nonplanar graph G, algorithm MAXIMAL-PLANARIZE constructs a maximal planar subgraph of G which contains G,,. This latter algorithm can also be used to maximally planarize a biconnected planar graph.