Given an undirected graph, the planarity testing problem is to determine whether the graph can be drawn in the plane without any crossing edges. Linear time planarity testing algorithms have previously been designed by Hopcroft and Tarjan, and by Booth and Lueker. However, their approaches are quite involved. Several other approaches have also been developed for simplifying the planariy test. In this paper, we developed a very simple linear time testing algorithm based only on a depth-first search tree. When the given graph is not planar, our algorithm immediately produces explicit Kuratowski's subgraphs. A new data structure, PC-trees, is introduced, which can be viewed as abstract subembeddings of actual planar embeddings. A graph-reduction technique is adopted so that the embeddings for the planar biconnected components constructed at each iteration never have to be changed. The recognition and embedding are actually done simultaneously in our algorithm 1 . The implementation of o...

. Given an undirected graph G, the maximal planar subgraph problem is to determine a planar subgraph H of G such that no edge of G-H can be added to H without destroying planarity. Polynomial algorithms have been obtained by Jakayumar, Thulasiraman and Swamy [6] and Wu [9]. O(mlogn) algorithms were previously given by Di Battista and Tamassia [3] and Cai, Han and Tarjan [2]. A recent O(m a a a a a a a a a a a a a a a a a a a a a a a a a (n)) algorithm was obtained by La Poute [7]. Our algorithm is based on a simple planarity test [5] developed by the author, which is a vertex addition algorithm based on a depth-first-search ordering. The planarity test [5] uses no complicated data structure and is conceptually simpler than Hopcroft and Tarjan's path addition and Lempel, Even and Cederbaum's vertex addition approaches. 1 1. Introduction Given an undirected graph, the planarity testing problem is to determine whether there exists a clockwise edge ordering around each vertex such that t...

In Shih & Hsu [9] a simpler planarity test was introduced utilizing a data structure called PC-trees (generalized from PQ-trees). In this paper we give an efficient implementation of that linear time algorithm and illustrate in detail how to obtain a Kuratowski subgraph when the given graph is not planar, and how to obtain the embedding alongside the testing algorithm. We have implemented the algorithm using LEDA and an object code is available at

not Available. Characterizing Proximity Trees Prosenjit Bose, William Lenhart, y and Giuseppe Liotta z Much attention has been given over the past several years to developing algorithms for embedding abstract graphs in the plane such that the resulting drawing has certain geometric properties. For example, those graphs which admit planar drawings have been completely characterized and efficient algorithms for producing planar drawings of these graphs have been designed ([4], [9]). For an overview of graph drawing problems and algorithms, the reader is referred to the excellent bibliography of Di Battista, Eades, Tamassia and Tollis [2]. Moreover, many problems in pattern recognition and classification, geographic variation analysis, geographic information systems, computational geometry, computational morphology, and computer vision use the underlying structure present in a set of data points revealed by means of a proximity graph. A proximity graph attempts to exhibit the rela...

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