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StraightLine Drawing Algorithms for Hierarchical Graphs and Clustered Graphs
 Algorithmica
, 1999
"... Hierarchical graphs and clustered graphs are useful nonclassical graph models for structured relational information. Hierarchical graphs are graphs with layering structures; clustered graphs are graphs with recursive clustering structures. Both have applications in CASE tools, software visualizatio ..."
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Cited by 59 (12 self)
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Hierarchical graphs and clustered graphs are useful nonclassical graph models for structured relational information. Hierarchical graphs are graphs with layering structures; clustered graphs are graphs with recursive clustering structures. Both have applications in CASE tools, software visualization, and VLSI design. Drawing algorithms for hierarchical graphs have been well investigated. However, the problem of straightline representation has not been solved completely. In this paper, we answer the question: does every planar hierarchical graph admit a planar straightline hierarchical drawing? We present an algorithm that constructs such drawings in linear time. Also, we answer a basic question for clustered graphs, that is, does every planar clustered graph admit a planar straightline drawing with clusters drawn as convex polygons? We provide a method for such drawings based on our algorithm for hierarchical graphs.
Width parameters beyond treewidth and their applications
 Computer Journal
, 2007
"... Besides the very successful concept of treewidth (see [Bodlaender, H. and Koster, A. (2007) Combinatorial optimisation on graphs of bounded treewidth. These are special issues on Parameterized Complexity]), many concepts and parameters measuring the similarity or dissimilarity of structures compare ..."
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Cited by 19 (0 self)
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Besides the very successful concept of treewidth (see [Bodlaender, H. and Koster, A. (2007) Combinatorial optimisation on graphs of bounded treewidth. These are special issues on Parameterized Complexity]), many concepts and parameters measuring the similarity or dissimilarity of structures compared to trees have been born and studied over the past years. These concepts and parameters have proved to be useful tools in many applications, especially in the design of efficient algorithms. Our presented novel look at the contemporary developments of these ‘width ’ parameters in combinatorial structures delivers—besides traditional treewidth and derived dynamic programming schemes—also a number of other useful parameters like branchwidth, rankwidth (cliquewidth) or hypertreewidth. In this contribution, we demonstrate how ‘width ’ parameters of graphs and generalized structures (such as matroids or hypergraphs), can be used to improve the design of parameterized algorithms and the structural analysis in other applications on an abstract level.
Linear Time Algorithm to Recognize Clustered Planar Graphs and its Parallelization
 98, 3rd Latin American symposium on theoretical informatics
, 1998
"... We develop a linear time algorithm for the following problem: Given a graph G and a hierarchical clustering of the vertices, such that all clusters induce connected subgraphs, determine whether G can be embedded into the plane, such that no cluster has a hole. This is an improvement to the O(n 2 )a ..."
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Cited by 14 (0 self)
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We develop a linear time algorithm for the following problem: Given a graph G and a hierarchical clustering of the vertices, such that all clusters induce connected subgraphs, determine whether G can be embedded into the plane, such that no cluster has a hole. This is an improvement to the O(n 2 )algorithm of Q.W. Feng et al. [6] and the algorithm of Lengauer [12].
Fixpoint logics on hierarchical structures
 Proceedings of the 25th International Conference on Foundations of Software Technology and Theoretical Computer Science (FSTTCS 2005), Hyderabad (Indien), number 3821 in Lecture Notes in Computer Science
, 2005
"... Abstract. Hierarchical graph definitions allow a modular description of graphs using modules for the specification of repeated substructures. Beside this modularity, hierarchical graph definitions allow to specify graphs of exponential size using polynomial size descriptions. In many cases, this suc ..."
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Cited by 6 (1 self)
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Abstract. Hierarchical graph definitions allow a modular description of graphs using modules for the specification of repeated substructures. Beside this modularity, hierarchical graph definitions allow to specify graphs of exponential size using polynomial size descriptions. In many cases, this succinctness increases the computational complexity of decision problems. In this paper, the modelchecking problem for the modal µcalculus and (monadic) least fixpoint logic on hierarchically defined graphs is investigated. In order to analyze the modal µcalculus, parity games on hierarchically defined graphs are studied. 1
ModelChecking Hierarchical Structures
, 2007
"... Hierarchical graph definitions allow a modular description of structures using modules for the specification of repeated substructures. Beside this modularity, hierarchical graph definitions allow to specify structures of exponential size using polynomial size descriptions. In many cases, this succi ..."
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Cited by 5 (1 self)
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Hierarchical graph definitions allow a modular description of structures using modules for the specification of repeated substructures. Beside this modularity, hierarchical graph definitions allow to specify structures of exponential size using polynomial size descriptions. In many cases, this succinctness increases the computational complexity of decision problems when input structures are defined hierarchically. In this paper, the modelchecking problem for firstorder logic (FO), monadic secondorder logic (MSO), and secondorder logic (SO) on hierarchically defined input structures is investigated. It is shown that in general these modelchecking problems are exponentially harder than their nonhierarchical counterparts, where the input structures are given explicitly. As a consequence, several new complete problems for the levels of the polynomial time hierarchy and the exponential time hierarchy are obtained. Based on classical results of Gaifman and Courcelle, two restrictions on the structure of hierarchical graph definitions that lead to more efficient modelchecking algorithms are presented.
On embedding a cycle in a plane graph
 PROC. 13TH INT. SYMP. GRAPH DRAWING (GD’05), VOLUME 3843 OF LNCS
, 2006
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CPlanarity of CConnected Clustered Graphs  Part I – Characterization
, 2006
"... We present a characterization of the cplanarity of cconnected clustered graphs. The characterization is based on the interplay between the hierarchy of the clusters and the hierarchy of the triconnected and biconnected components of the graph underlying the clustered graph. In a companion paper [2 ..."
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Cited by 2 (1 self)
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We present a characterization of the cplanarity of cconnected clustered graphs. The characterization is based on the interplay between the hierarchy of the clusters and the hierarchy of the triconnected and biconnected components of the graph underlying the clustered graph. In a companion paper [2] we exploit such a characterization to give a linear time cplanarity testing and embedding algorithm.
Linear Algebra Using Maple's LargeExpressions Package
, 2006
"... The package LargeExpressions has been available in Maple for a number of years, but it is not well known. It provides tools for managing large expressions. In this paper, we describe a new application of this tool to the LU factoring of matrices. We describe a function that factors a matrix and ..."
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Cited by 1 (0 self)
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The package LargeExpressions has been available in Maple for a number of years, but it is not well known. It provides tools for managing large expressions. In this paper, we describe a new application of this tool to the LU factoring of matrices. We describe a function that factors a matrix and expresses the results using a hierarchical representation. As part of the LU factoring, we introduce several strategies for pivoting, veiling an expression and zerorecognition in our function. All these strategies can be chosen based on the application.
CPlanarity of cconnected clustered graphs
, 2008
"... We present the first characterization of cplanarity for cconnected clustered graphs. The characterization is based on the interplay between the hierarchy of the clusters and the hierarchies of the triconnected and biconnected components of the underlying graph. Based on such a characterization, we ..."
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Cited by 1 (1 self)
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We present the first characterization of cplanarity for cconnected clustered graphs. The characterization is based on the interplay between the hierarchy of the clusters and the hierarchies of the triconnected and biconnected components of the underlying graph. Based on such a characterization, we provide a lineartime cplanarity testing and embedding algorithm for cconnected clustered graphs. The algorithm is reasonably easy to implement, since it exploits as building blocks simple algorithmic tools like the computation of lowest common ancestors, minimum and maximum spanning trees, and counting sorts. It also makes use of wellknown data structures as SPQRtrees and BCtrees. If the test fails, the algorithm identifies a structural element responsible for the noncplanarity of the input clustered graph.