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30
A FixedParameter Approach to 2Layer Planarization
, 2006
"... A bipartite graph is biplanar if the vertices can be placed on two parallel lines (layers) inthe plane such that there are no edge crossings when edges are drawn as line segments between the layers. In this paper we study the 2LAYER PLANARIZATION problem: Can k edges be deleted from a given graph ..."
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Cited by 8 (2 self)
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A bipartite graph is biplanar if the vertices can be placed on two parallel lines (layers) inthe plane such that there are no edge crossings when edges are drawn as line segments between the layers. In this paper we study the 2LAYER PLANARIZATION problem: Can k edges be deleted from a given graph G so that the remaining graph is biplanar? This problem is NPcomplete, and remains so if the permutation of the vertices in one layer is fixed (the 1LAYER PLANARIZATION problem). We prove that these problems are fixedparameter tractable by giving lineartime algorithms for their solution (for fixed k). In particular, we solve the 2LAYER PLANARIZATION problem in O(k · 6 k +G) time and the 1LAYER PLANARIZATION problem in O(3 k ·G) time. We also show that there are polynomialtime constantapproximation algorithms for both problems.
Nice Drawings for Planar Bipartite Graphs
 In 3rd Italian Conference on Algorithms and Complexity, CIAC '97
, 1997
"... Graph drawing algorithms usually attempt to display the characteristic properties of the input graphs. In this paper we consider the class of planar bipartite graphs and try to achieve planar drawings such that the bipartiteness property is cleary shown. To this aim, we develop several models, give ..."
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Cited by 8 (0 self)
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Graph drawing algorithms usually attempt to display the characteristic properties of the input graphs. In this paper we consider the class of planar bipartite graphs and try to achieve planar drawings such that the bipartiteness property is cleary shown. To this aim, we develop several models, give efficient algorithms to find a corresponding drawing if possible or prove the hardness of the problem. 1 Introduction Graph drawing is a more and more developing method to visualize data and their relations. The main goal is to draw the graph in such a way that certain properties are clearly displayed: Planar graphs should be drawn planar [6], symmetries should be displayed [3, 11], if the graph is directed and acyclic then it should be drawn 'upward' [12], cliques should be easily recognized. There are many more properties developed in graph theory and graph algorithms that are worth to be displayed [1]. Another important property is the bipartiteness. A graph G = (V; E) with a set V of v...
Drawing planar partitions III: Two constrained embedding problems
, 1998
"... In two previous papers, we studied the following problem: Given a planar graph G = (V; E) and a partition V = A [ B of the vertices. Can we draw G without crossing such that the partition is clearly visible? For three models used to display the partition, we developed necessary and sufficient con ..."
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Cited by 8 (1 self)
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In two previous papers, we studied the following problem: Given a planar graph G = (V; E) and a partition V = A [ B of the vertices. Can we draw G without crossing such that the partition is clearly visible? For three models used to display the partition, we developed necessary and sufficient conditions for the existence of such a drawing. The time to test these conditions was dominated by verifying whether there exists a planar embedding of G that satisfies certain additional properties. In this paper, we study how these constrained embedding tests can be done in linear time.
Pathwidth and Layered Drawings of Trees
 INTERNAT. J. COMPUT. GEOM. APPL
, 2002
"... An hlayer drawing of a graph G is a planar drawing of G in which each vertex is placed on one of h parallel lines and each edge is drawn as a straight line between its endvertices. In such a drawing, we say that an edge is proper if its endpoints lie on adjacent layers, flat if they lie on the ..."
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Cited by 7 (3 self)
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An hlayer drawing of a graph G is a planar drawing of G in which each vertex is placed on one of h parallel lines and each edge is drawn as a straight line between its endvertices. In such a drawing, we say that an edge is proper if its endpoints lie on adjacent layers, flat if they lie on the same layer and long otherwise. Thus, a proper hlayer drawing contains only proper edges, a short hlayer drawing contains no long edges, an upright hlayer drawing contains no flat edges, and an unconstrained hlayer drawing contains any type of edge. We prove
Drawing Graphs on Two and Three Lines
 GRAPH DRAWING, 10TH INTERNATIONAL SYMPOSIUM (GD 2002), VOLUME TO APPEAR OF LECTURE NOTES IN COMPUTER SCIENCE
, 2002
"... We give a lineartime algorithm to decide whether a graph has a planar LLdrawing, i.e. a planar drawingo two parallel lines. This has previo)L/ beenkno wnoLq fo trees. We utilize this resultto oult planar drawings on three lines for a generalization of bipartite graphs, also in linear time. ..."
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Cited by 7 (1 self)
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We give a lineartime algorithm to decide whether a graph has a planar LLdrawing, i.e. a planar drawingo two parallel lines. This has previo)L/ beenkno wnoLq fo trees. We utilize this resultto oult planar drawings on three lines for a generalization of bipartite graphs, also in linear time.
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...
New Bounds on the Barycenter Heuristic for Bipartite Graph Drawing
 INFORMATION PROCESSING LETTERS
, 2001
"... The barycenter heuristic is often used in practice to solve the NPhard twolayer edge crossing minimization problem. It is wellknown that the barycenter heuristic can give solutions as bad as Ω(√n) times the optimum, where n is the number of nodes in the graph. However, the example used in the ..."
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Cited by 5 (0 self)
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The barycenter heuristic is often used in practice to solve the NPhard twolayer edge crossing minimization problem. It is wellknown that the barycenter heuristic can give solutions as bad as Ω(√n) times the optimum, where n is the number of nodes in the graph. However, the example used in the proof has many isolated nodes. Makinen [8] conjectured that a better ratio bound is possible if isolated nodes are not present. We show that the ratio bound for the barycenter heuristic is still Ω(√n) even for connected bipartite graphs. We also prove a tight constant ratio bound for the barycenter heuristic on boundeddegree graphs. The bound is d  1, where d is the maximum degree of a node in the layer that can be permuted.
Experiments with the fixedparameter approach for twolayer planarization
 In [15
, 2003
"... Abstract. We present computational results of an implementation based on the fixed parameter tractability (FPT) approach for biplanarizing graphs. These results show that the implementation can efficiently minimum biplanarizing sets containing up to about 18 edges, thus making it comparable to previ ..."
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Cited by 4 (2 self)
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Abstract. We present computational results of an implementation based on the fixed parameter tractability (FPT) approach for biplanarizing graphs. These results show that the implementation can efficiently minimum biplanarizing sets containing up to about 18 edges, thus making it comparable to previous integer linear programming approaches. We show how our implementation slightly improves the theoretical running time to O(6 bpr (G) + G). Finally, we explain how our experimental work predicts how performance on sparse graphs may be improved. 1
Graph Visualization in Software Analysis
 Proceedings of 1992 Symposium on Assessment of Quality Software Development Tools
, 1992
"... Directed graphs are ubiquitous in most aspects of software analysis. Presented abstractly, as a list of edges, a graph does not manifest much of the important structural information that becomes obvious if the graph displayed pictorially. This paper presents a technique for drawing directed graphs q ..."
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Cited by 4 (1 self)
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Directed graphs are ubiquitous in most aspects of software analysis. Presented abstractly, as a list of edges, a graph does not manifest much of the important structural information that becomes obvious if the graph displayed pictorially. This paper presents a technique for drawing directed graphs quickly and attractively. It also describes how a tool implementing this technique has been used, in conjunction with other programming and analysis tools, in various aspects of software engineering. 1
Characterization of Level NonPlanar Graphs by Minimal Patterns
 PROC. COMPUTING AND COMBINATORICS, COCOON 2000, VOLUME 1858 OF LNCS
, 2000
"... A level graph G = (V; E; ) is a directed acyclic graph with a mapping : V ! f1; 2; : : : ; kg, k 1, that partitions the vertex set V as V = V 1 [V 2 [: : :[V k , V j = 1 (j), V i \V j = ; for i 6= j, such that (v) = (u)+1 for each edge (u; v) 2 E. The graph G is level planar if it can be drawn in ..."
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Cited by 4 (0 self)
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A level graph G = (V; E; ) is a directed acyclic graph with a mapping : V ! f1; 2; : : : ; kg, k 1, that partitions the vertex set V as V = V 1 [V 2 [: : :[V k , V j = 1 (j), V i \V j = ; for i 6= j, such that (v) = (u)+1 for each edge (u; v) 2 E. The graph G is level planar if it 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 i) j x 2 Rg, the edges are drawn monotonically with respect to the vertical direction, and no edges intersect except at their end vertices. In this paper we give a characterization of level planar graphs in terms of minimal forbidden subgraphs called minimal level nonplanar subgraph patterns (MLNP). We show that a MLNP is completely characterized by either a tree, a level nonplanar cycle or a level planar cycle with certain path augmentations. These characterizations are an important first step towards attacking the NPhard level planarization problem.