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A New Planarity Test
, 1999
"... 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 ..."
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Cited by 22 (2 self)
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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 depthfirst search tree. When the given graph is not planar, our algorithm immediately produces explicit Kuratowski's subgraphs. A new data structure, PCtrees, is introduced, which can be viewed as abstract subembeddings of actual planar embeddings. A graphreduction 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...
Competitive Robot Mapping with Homogeneous Markers
 IEEE Trans. on Robotics and Automation
, 1996
"... : We consider the robot exploration problem of graph maps with homogeneous markers, following the graph world model introduced by Dudek et al. [DJMW]. The environment is a graph consisting of nodes and edges, where the robot can navigate from one node to another through an edge connecting these two ..."
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Cited by 21 (3 self)
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: We consider the robot exploration problem of graph maps with homogeneous markers, following the graph world model introduced by Dudek et al. [DJMW]. The environment is a graph consisting of nodes and edges, where the robot can navigate from one node to another through an edge connecting these two nodes. However, the robot may not distinguish one node (or edge) from another in this unknown graph. All the nodes (edges) look the same. However, at each node, the robot can observe a consistent local relative orientation of its incident edges, that is, a cyclic order of edges incident to the node. To assist the robot's task of mapping the environment, it can put homogeneous (i.e., identical) marks on nodes or edges which can be recognized later. The total number of edges traversed when constructing a map of the graph is often used as a performance measure for robot strategies. However, since the graph is unknown, a strategy may be efficient in one situation but not in others. Thus, there i...
A simple test for planar graphs
 Proceedings of the International Workshop on Discrete Mathematics and Algorithms, University of Hong Kong
, 1993
"... 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 ..."
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Cited by 7 (3 self)
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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. We develop a very simple linear time testing algorithm based only on a depthfirst search tree. Our algorithm uses Kuratowski's theorem very explicitly. A graphreduction technique is adopted so that the embeddings for the planar biconnected components constructed at each iteration never have to be changed. 1.
A linear time algorithm for finding a maximal planar subgraph based on PCtrees
 of Lecture Notes in Computer Science
, 2005
"... ABSTRACT. Given an undirected graph G, the maximal planar subgraph problem is to determine a planar subgraph H of G such that no edge of GH can be added to H without destroying planarity. Polynomial algorithms have been obtained by Jakayumar, Thulasiraman and Swamy [6] and Wu [9]. O(mlogn) algorith ..."
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Cited by 5 (0 self)
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ABSTRACT. Given an undirected graph G, the maximal planar subgraph problem is to determine a planar subgraph H of G such that no edge of GH 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(mα (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 depthfirstsearch 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.
An efficient implementation of the PCtrees algorithm of shih and hsu’s planarity test
 Institute of Information Science, Academia Sinica
, 2003
"... In Shih & Hsu [9] a simpler planarity test was introduced utilizing a data structure called PCtrees (generalized from PQtrees). 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 n ..."
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In Shih & Hsu [9] a simpler planarity test was introduced utilizing a data structure called PCtrees (generalized from PQtrees). 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
Periodic assignment and graph colouring
 Discrete Appl. Math
, 1994
"... We analyse the problem of executing periodic operations on a minimum number of identical processors under different constraints. The analysis is based on a reformulation of the problem in terms of graph colouring. It is shown that different constraints result in colouring problems that are defined o ..."
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We analyse the problem of executing periodic operations on a minimum number of identical processors under different constraints. The analysis is based on a reformulation of the problem in terms of graph colouring. It is shown that different constraints result in colouring problems that are defined on different classes of graphs, viz. interval graphs, circulararc graphs and periodicinterval graphs. We discuss the colouring of such graphs in detail
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The author has granted a non L'auteur a accordé une licence non exclusive licence allowing the exclusive permettant à la National Library of Canada to ~ibliothe~ue nationale du Canada de reproduce, loan, distriiute or seil reproduire, prêter, distribuer ou copies of this thesis in microforrn, vendre des copies de cette thèse sous paper or electronic formats. la forme de microfiche/nlm, de reproduction sw papier ou sur format électronique. The author retains ownership of the L'auteur conserve la propriété du copyxight in this thesis. Neither the droit d'auteur qui protège cette thèse. thesis nor substaxitial extracts fiom it Ni la thèse ni des extraits substantiels may be printed or otherwise de celleci ne doivent être imprimés reproduced without the author's ou autrement reproduits sans son permission. autorisation.