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On the cutting edge: Simplified O(n) planarity by edge addition
 Journal of Graph Algorithms and Applications
, 2004
"... www.cs.uvic.ca/˜wendym ..."
ONLINE PLANARITY TESTING
, 1996
"... The online planaritytesting problem consists of performing the following operations on a planar graph G: (i) testing if a new edge can be added to G so that the resulting graph is itself planar; (ii) adding vertices and edges such that planarity is preserved. An efficient technique for online plan ..."
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Cited by 26 (2 self)
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The online planaritytesting problem consists of performing the following operations on a planar graph G: (i) testing if a new edge can be added to G so that the resulting graph is itself planar; (ii) adding vertices and edges such that planarity is preserved. An efficient technique for online planarity testing of a graph is presented that uses O(n) space and supports tests and insertions of vertices and edges in O(log n) time, where n is the current number of vertices of G. The bounds for tests and vertex insertions are worstcase and the bound for edge insertions is amortized. We also present other applications of this technique to dynamic algorithms for planar graphs.
Algorithms for Drawing Clustered Graphs
, 1997
"... In the mid 1980s, graphics workstations became the main platforms for software and information engineers. Since then, visualization of relational information has become an essential element of software systems. Graphs are commonly used to model relational information. They are depicted on a graphics ..."
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Cited by 25 (2 self)
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In the mid 1980s, graphics workstations became the main platforms for software and information engineers. Since then, visualization of relational information has become an essential element of software systems. Graphs are commonly used to model relational information. They are depicted on a graphics workstation as graph drawings. The usefulness of the relational model depends on whether the graph drawings effectively convey the relational information to the users. This thesis is concerned with finding good drawings of graphs. As the amount of information that we want to visualize becomes larger and the relations become more complex, the classical graph model tends to be inadequate. Many extended models use a node hierarchy to help cope with the complexity. This thesis introduces a new graph model called the clustered graph. The central theme of the thesis is an investigation of efficient algorithms to produce good drawings for clustered graphs. Although the criteria for judging the qua...
Stop Minding Your P's and Q's: A Simplified O(n) Planar Embedding Algorithm
 In Proc. 10th ACMSIAM Symposium on Discrete Algorithms, SODA
, 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 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 comp ..."
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Cited by 25 (4 self)
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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 first stnumber 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 PQtree. The Pnodes in a PQtree represent parts of the partially embedded graph that can be permuted, and the Qnodes represent parts that can be flipped. We avoid the use of Pnodes by not connecting pieces together until they become biconnected. We avoid Q nodes by using a data structure which allows biconnected components to be flipped in O(1) time. 1 In...
An InformationTheoretic Upper Bound of Planar Graphs Using Triangulation
, 2003
"... We propose a new linear time algorithm to represent a planar graph. Based on a specific triangulation of the graph, our coding takes on average 5.03 bits per node, and 3.37 bits per node if the graph is maximal. We derive from this representation that the number of unlabeled planar graphs with n ..."
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Cited by 24 (5 self)
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We propose a new linear time algorithm to represent a planar graph. Based on a specific triangulation of the graph, our coding takes on average 5.03 bits per node, and 3.37 bits per node if the graph is maximal. We derive from this representation that the number of unlabeled planar graphs with n nodes is at most 2 n+O(log n) where 5.007. The current lower bound is 2 n+(log n) for 4.71. We also show that almost all unlabeled and almost all labeled nnode planar graphs have at least 1.70n edges and at most 2.54n edges.
Parallel transitive closure and point location in planar structures
 SIAM J. COMPUT
, 1991
"... Parallel algorithms for several graph and geometric problems are presented, including transitive closure and topological sorting in planar stgraphs, preprocessing planar subdivisions for point location queries, and construction of visibility representations and drawings of planar graphs. Most of th ..."
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Cited by 23 (11 self)
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Parallel algorithms for several graph and geometric problems are presented, including transitive closure and topological sorting in planar stgraphs, preprocessing planar subdivisions for point location queries, and construction of visibility representations and drawings of planar graphs. Most of these algorithms achieve optimal O(log n) running time using n = log n processors in the EREW PRAM model, n being the number of vertices.
Level Planarity Testing in Linear Time
, 1999
"... A level graph G = (V; E; OE) is a directed acyclic graph with a mapping OE : V ! f1; 2; : : : ; kg, k 1, that partitions the vertex set V as V = V 1 [V 2 [ \Delta \Delta \Delta [V k , V j = OE \Gamma1 (j), V i " V j = ; for i 6= j, such that OE(v) OE(u) + 1 for each edge (u; v) 2 E. The level planar ..."
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Cited by 21 (2 self)
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A level graph G = (V; E; OE) is a directed acyclic graph with a mapping OE : V ! f1; 2; : : : ; kg, k 1, that partitions the vertex set V as V = V 1 [V 2 [ \Delta \Delta \Delta [V k , V j = OE \Gamma1 (j), V i " V j = ; for i 6= j, such that OE(v) OE(u) + 1 for each edge (u; v) 2 E. The level planarity testing problem is to decide if G can be drawn in the plane such that for each level V i , all v 2 V i are drawn on the line l i = f(x; k \Gamma i) j x 2 Rg, the edges are drawn monotone with respect to the vertical direction, and no edges intersect except at their end vertices. If G has a single source, the test can be performed in O(jV j) time by an algorithm of Di Battista and Nardelli [1988] that uses the PQtree data structure introduced by Booth and Lueker [1976]. PQtrees have also been proposed by Heath and Pemmaraju [1995, 1996] to test level planarity of level directed acyclic graphs with several sources and sinks. It has been shown in Jünger, Leipert, and...
Level Planar Embedding in Linear Time
, 1999
"... A level graph G  (V, E, q) is a directed acyclic graph with a mapping q: V  {1, 2,...,k), k _ 1, that partitions the vertex set V as V V10V20 ...V k, vj = ql(j), Vi [ vj = for i j, such that q(v) _ q(u) + 1 for each edge (u, v) E. The level planarity testing problem is to decide if G can be ..."
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Cited by 20 (0 self)
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A level graph G  (V, E, q) is a directed acyclic graph with a mapping q: V  {1, 2,...,k), k _ 1, that partitions the vertex set V as V V10V20 ...V k, vj = ql(j), Vi [ vj = for i j, such that q(v) _ q(u) + 1 for each edge (u, v) E. The level planarity testing problem is to decide if G can be drawn in the plane such that for each level V i, all v V i are drawn on the line li  {(x, k  i) ] x ), the edges are drawn monotonically with respect to the vertical direction, and no edges intersect except at their end vertices. In order to
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 20 (2 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 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 19 (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...