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Cplanarity of extrovert clustered graphs
 In Graph Drawing
, 2005
"... Abstract. A clustered graph has its vertices grouped into clusters in a hierarchical way via subset inclusion, thereby imposing a tree structure on the clustering relationship. The cplanarity problem is to determine if such a graph can be drawn in a planar way, with clusters drawn as nested regions ..."
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Abstract. A clustered graph has its vertices grouped into clusters in a hierarchical way via subset inclusion, thereby imposing a tree structure on the clustering relationship. The cplanarity problem is to determine if such a graph can be drawn in a planar way, with clusters drawn as nested regions and with each edge (drawn as a curve between vertex points) crossing the boundary of each region at most once. Unfortunately, as with the graph isomorphism problem, it is open as to whether the cplanarity problem is NPcomplete or in P. In this paper, we show how to solve the cplanarity problem in polynomial time for a new class of clustered graphs, which we call extrovert clustered graphs. This class is quite natural (we argue that it captures many clustering relationships that are likely to arise in practice) and includes the clustered graphs tested in previous work by Dahlhaus, as well as Feng, Eades, and Cohen. Interestingly, this class of graphs does not include, nor is it included by, a class studied recently by Gutwenger et al.; therefore, this paper offers an alternative advancement in our understanding of the efficient drawability of clustered graphs in a planar way. Our testing algorithm runs in O(n 3) time and implies an embedding algorithm with the same time complexity. 1
Fully Dynamic Planarity Testing with Applications
"... The fully dynamic planarity testing problem consists of performing an arbitrary sequence of the following three kinds of operations on a planar graph G: (i) insert an edge if the resultant graph remains planar; (ii) delete an edge; and (iii) test whether an edge could be added to the graph without ..."
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Cited by 6 (0 self)
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The fully dynamic planarity testing problem consists of performing an arbitrary sequence of the following three kinds of operations on a planar graph G: (i) insert an edge if the resultant graph remains planar; (ii) delete an edge; and (iii) test whether an edge could be added to the graph without violating planarity. We show how to support each of the above operations in O(n2=3) time, where n is the number of vertices in the graph. The bound for tests and deletions is worstcase, while the bound for insertions is amortized. This is the first algorithm for this problem with sublinear running time, and it affirmatively answers a question posed in [11]. The same data structure has further applications in maintaining the biconnected and triconnected components of a dynamic planar graph. The time bounds are the same: O(n2=3) worstcase time per edge deletion, O(n2=3) amortized time per edge insertion, and O(n2=3) worstcase time to check whether two vertices are either biconnected or triconnected.
A Note on Computing a Maximal Planar Subgraph using PQTrees
, 1998
"... The problem of computing a maximal planar subgraph of a non planar graph has been deeply investigated over the last 20 years. Several attempts have been tried to solve the problem with the help of PQtrees. The latest attempt has been reported by Jayakumar et al. [10]. In this paper we show that ..."
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Cited by 6 (3 self)
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The problem of computing a maximal planar subgraph of a non planar graph has been deeply investigated over the last 20 years. Several attempts have been tried to solve the problem with the help of PQtrees. The latest attempt has been reported by Jayakumar et al. [10]. In this paper we show that the algorithm presented by Jayakumar et al. is not correct. We show that it does not necessarily compute a maximal planar subgraph and we note that the same holds for a modified version of the algorithm presented by Kant [12]. Our conclusions most likely suggest not to use PQtrees at all for this specific problem.
IMPROVED COMPACT VISIBILITY REPRESENTATION OF Planar Graph via Schnyder’s Realizer
 SIAM J. DISCRETE MATH. C ○ 2004 SOCIETY FOR INDUSTRIAL AND APPLIED MATHEMATICS VOL. 18, NO. 1, PP. 19–29
, 2004
"... Let G be an nnode planar graph. In a visibility representation of G,eachnodeofG is represented by a horizontal line segment such that the line segments representing any two adjacent nodes of G are vertically visible to each other. In the present paper we give the best known compact visibility repre ..."
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Cited by 6 (1 self)
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Let G be an nnode planar graph. In a visibility representation of G,eachnodeofG is represented by a horizontal line segment such that the line segments representing any two adjacent nodes of G are vertically visible to each other. In the present paper we give the best known compact visibility representation of G. Given a canonical ordering of the triangulated G, our algorithm draws the graph incrementally in a greedy manner. We show that one of three canonical orderings obtained �from Schnyder’s � realizer for the triangulated G yields a visibility representation of G no wider than 22n−40. Our easytoimplement O(n)time algorithm bypasses the complicated subroutines for 15 fourconnected components and fourblock trees required by the best previously known algorithm of Kant. Our result provides a negative answer to Kant’s open question about whether � � 3n−6 is a 2 worstcase lower bound on the required width. Also, if G has no degreethree (respectively, degreefive) internal node, then our visibility representation for G is no wider than � �
Balanced VertexOrderings of Graphs
, 2002
"... We consider the problem of determining a balanced ordering of the vertices of a graph; that is, the neighbors of each vertex v are as evenly distributed to the left and right of v as possible. This problem, which has applications in graph drawing for example, is shown to be NPhard, and remains N ..."
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Cited by 5 (4 self)
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We consider the problem of determining a balanced ordering of the vertices of a graph; that is, the neighbors of each vertex v are as evenly distributed to the left and right of v as possible. This problem, which has applications in graph drawing for example, is shown to be NPhard, and remains NPhard for bipartite simple graphs with maximum degree six. We then describe and analyze a number of methods for determining a balanced vertexordering, obtaining optimal orderings for directed acyclic graphs and graphs with maximum degree three. Finally we
Clustered Graphs and Cplanarity
 In 3rd Annual European Symposium on Algorithms (ESA’95), LNCS 979
, 1995
"... In this paper, we introduce a new graph model known as clustered graphs, i.e. graphs with recursive clustering structures. This graph model has many applications in informational and mathematical sciences. In particular, we study Cplanarity of clustered graphs. Given a clustered graph, the Cplanar ..."
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Cited by 5 (2 self)
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In this paper, we introduce a new graph model known as clustered graphs, i.e. graphs with recursive clustering structures. This graph model has many applications in informational and mathematical sciences. In particular, we study Cplanarity of clustered graphs. Given a clustered graph, the Cplanarity testing problem is to determine whether the clustered graph can be drawn without edge crossings, or edgeregion crossings. In this paper, we present efficient algorithms for testing Cplanarity and finding Cplanar embeddings of clustered graphs. 1 Introduction Representing information visually, or by drawing graphs can greatly improve the effectiveness of user interfaces in many relational information systems [12, 17, 18, 5]. Developing algorithms for drawing graphs automatically and efficiently has become the interest of research for many computer scientists. Research in this area has been very active for the last decade. A recent survey citelabel13new of literature in this area inclu...
TwoBend ThreeDimensional Orthogonal Grid Drawing of Maximum Degree Five Graphs
 IEEE ISSCC 94; Pennsylvania
, 1998
"... Some recent algorithms for 3dimensional orthogonal graph drawing use no more than 3 bends per edge route. It is unknown if there exists a graph requiring a 3bend edge route. In this paper we present an algorithm for 2bend 3dimensional orthogonal grid drawing of maximum degree 5 graphs. In additi ..."
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Cited by 5 (1 self)
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Some recent algorithms for 3dimensional orthogonal graph drawing use no more than 3 bends per edge route. It is unknown if there exists a graph requiring a 3bend edge route. In this paper we present an algorithm for 2bend 3dimensional orthogonal grid drawing of maximum degree 5 graphs. In addition 2bend 3dimensional grid drawings of the 6regular multipartite graphs are given. 1 Introduction The investigation of graph drawing in the plane has been extensive (see [5] for a bibliographic survey). Prompted by advances in graphics workstations and applications including VLSI circuit design [17, 19] and software engineering [11, 18] there has been recent interest in graph visualisation in 3dimensional space. The 3dimensional orthogonal grid consists of grid points in 3dimensional space with integer coordinates, together with the axisparallel grid lines determined by these points. An orthogonal grid drawing of a graph places the vertices at grid points and routes the edges along...
Orthogonal Grid Drawing of Clustered Graphs
, 1996
"... Clustered graphs are graphs with recursive clustering structures over the vertices. For graphical representation, the clustering structure is represented by a simple region that contains the drawing of all the vertices which belong to that cluster. In this paper, we present an algorithm which pro ..."
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Cited by 4 (2 self)
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Clustered graphs are graphs with recursive clustering structures over the vertices. For graphical representation, the clustering structure is represented by a simple region that contains the drawing of all the vertices which belong to that cluster. In this paper, we present an algorithm which produces planar drawings of clustered graphs in a convention known as orthogonalgrid rectangular cluster drawings. The drawing produced by the algorithm has constant number of bends on each edge and has O(n 2 ) area, which is as good as existing results for classical graph drawings. 1 Introduction Clustered graphs are graphs with recursive clustering structures over the vertices (see Fig. 1). This type of clustering structure appears in many systems. Examples include CASE tools [19], management information systems [10], and VLSI design tools [8]. For graphical representation, the clustering structure is represented by a simple region that contains the drawing of all the vertices which ...
On Computing a Maximal Planar Subgraph using PQTrees
, 1996
"... The problem of computing a maximal planar subgraph of a nonplanar graph has been deeply investigated over the last 20 years. Several attempts have been tried to solve the problem with the help of PQtrees. The latest attempt has been reported by Jayakumar et al. (1989). In this paper we show that t ..."
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Cited by 4 (3 self)
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The problem of computing a maximal planar subgraph of a nonplanar graph has been deeply investigated over the last 20 years. Several attempts have been tried to solve the problem with the help of PQtrees. The latest attempt has been reported by Jayakumar et al. (1989). In this paper we show that the algorithm presented by Jayakumar et al. is not correct. We show that it does not necessarily compute a maximal planar subgraph and that the same holds for a modified version of the algorithm presented by Kant (1992). Our conclusions most likely suggest not to use PQtrees at all for this specific problem.
A linear algorithm for finding a maximal planar subgraph
 SIAM J. Disc. Math
, 2006
"... Abstract. We construct an optimal lineartime algorithm for the maximal planar subgraph problem: given a graph G, find a planar subgraph G ′ of G such that adding to G ′ an extra edge of G results in a nonplanar graph. Our solution is based on a fast data structure for incremental planarity testing ..."
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Cited by 4 (0 self)
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Abstract. We construct an optimal lineartime algorithm for the maximal planar subgraph problem: given a graph G, find a planar subgraph G ′ of G such that adding to G ′ an extra edge of G results in a nonplanar graph. Our solution is based on a fast data structure for incremental planarity testing of triconnected graphs and a dynamic graph search procedure. Our algorithm can be transformed into a new optimal planarity testing algorithm. Key words. Planar graphs, planarity testing, incremental algorithms, graph planarization, data structures, triconnectivity. AMS subject classifications. 05C10, 05C85, 68R10, 68Q25, 68W40 1. Introduction. Agraphisplanar