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Testing Planarity of Partially Embedded Graphs
, 2009
"... We study the following problem: Given a planar graph G and a planar drawing (embedding) of a subgraph of G, can such a drawing be extended to a planar drawing of the entire graph G? This problem fits the paradigm of extending a partial solution to a complete one, which has been studied before in man ..."
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Cited by 23 (10 self)
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We study the following problem: Given a planar graph G and a planar drawing (embedding) of a subgraph of G, can such a drawing be extended to a planar drawing of the entire graph G? This problem fits the paradigm of extending a partial solution to a complete one, which has been studied before in many different settings. Unlike many cases, in which the presence of a partial solution in the input makes hard an otherwise easy problem, we show that the planarity question remains polynomialtime solvable. Our algorithm is based on several combinatorial lemmata which show that the planarity of partially embedded graphs meets the “oncas” behaviour – obvious necessary conditions for planarity are also sufficient. These conditions are expressed in terms of the interplay between (a) rotation schemes and containment relationships between cycles and (b) the decomposition of a graph into its connected, biconnected, and triconnected components. This implies that no dynamic programming is needed for a decision algorithm and that the elements of the decomposition can be processed independently. Further, by equipping the components of the decomposition with suitable data structures and by carefully splitting the problem into simpler subproblems, we improve our algorithm to reach lineartime complexity. Finally, we consider several generalizations of the problem, e.g. minimizing the number of edges of the partial embedding that need to be rerouted to extend it, and argue that they are NPhard. Also, we show how our algorithm can be applied to solve related Graph Drawing problems.
An SPQRTree Approach to Decide Special Cases of Simultaneous Embedding with Fixed Edges
, 2008
"... We present a lineartime algorithm for solving the simultaneous embedding problem with fixed edges (SEFE) for a planar graph and a pseudoforest (a graph with at most one cycle) by reducing it to the following embedding problem: Given a planar graph G, a cycle C of G, and a partitioning of the remai ..."
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Cited by 5 (0 self)
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We present a lineartime algorithm for solving the simultaneous embedding problem with fixed edges (SEFE) for a planar graph and a pseudoforest (a graph with at most one cycle) by reducing it to the following embedding problem: Given a planar graph G, a cycle C of G, and a partitioning of the remaining vertices of G, does there exist a planar embedding in which the induced subgraph on each vertex partite of G \ C is contained entirely inside or outside C? For the latter problem, we present an algorithm that is based on SPQRtrees and has linear running time. We also show how we can employ SPQRtrees to decide SEFE for two planar graphs where one graph has at most two cycles and the intersection is a pseudoforest in linear time. These results give rise to our hope that our SPQRtree approach might eventually lead to a polynomialtime algorithm for deciding the general SEFE problem for two planar graphs.
Upward Planarity Testing via SAT
"... A directed acyclic graph is upward planar if it allows a drawing without edge crossings where all edges are drawn as curves with monotonously increasing ycoordinates. The problem to decide whether a graph is upward planar or not is NPcomplete in general, and while special graph classes are polyn ..."
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Cited by 5 (1 self)
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A directed acyclic graph is upward planar if it allows a drawing without edge crossings where all edges are drawn as curves with monotonously increasing ycoordinates. The problem to decide whether a graph is upward planar or not is NPcomplete in general, and while special graph classes are polynomial time solvable, there is not much known about solving the problem for general graphs in practice. The only attempt so far was a branchandbound algorithm over the graph’s triconnectivity structure which was able to solve sparse graphs. In this paper, we propose a fundamentally different approach, based on the seemingly novel concept of ordered embeddings. We carefully model the problem as a special SAT instance, i.e., a logic formula for which we check satisfiability. Solving these SAT instances allows us to decide upward planarity for arbitrary graphs. We then show experimentally that this approach seems to dominate the known alternative approaches and is able to solve traditionally used graph drawing benchmarks effectively.
Visual Analysis of OnetoMany Matched Graphs
, 2010
"... Motivated by applications of social network analysis and of Web search clustering engines, we describe an algorithm and a system for the display and the visual analysis of two graphs G1 and G2 such that each Gi is defined on a different data set with its own primary relationships and there are secon ..."
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Cited by 2 (0 self)
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Motivated by applications of social network analysis and of Web search clustering engines, we describe an algorithm and a system for the display and the visual analysis of two graphs G1 and G2 such that each Gi is defined on a different data set with its own primary relationships and there are secondary relationships between the vertices of G1 and those of G2. Our main goal is to compute a drawing of G1 and G2 that makes clearly visible the relations between the two graphs by avoiding their crossings, and that also takes into account some other important aesthetic requirements like number of bends, area, and aspect ratio. Application examples and experiments on the system performances are also presented.
Crossing minimization and layouts of . . .
"... Many practical applications for drawing graphs are modeled by directed graphs with domain specific constraints. In this paper, we consider the problem of drawing directed hypergraphs with (and without) port constraints, which cover multiple realworld graph drawing applications like data flow diagr ..."
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Many practical applications for drawing graphs are modeled by directed graphs with domain specific constraints. In this paper, we consider the problem of drawing directed hypergraphs with (and without) port constraints, which cover multiple realworld graph drawing applications like data flow diagrams and electric schematics. Most existing algorithms for drawing hypergraphs with port constraints are adaptions of the framework originally proposed by Sugiyama et al. in 1981 for simple directed graphs. Recently, a practical approach for upward crossing minimization of directed graphs based on the planarization method was proposed [7]. With respect to the number of arc crossings, it clearly outperforms prior (mostly layeringbased) approaches. We show how to adopt this idea for hypergraphs with given port constraints, obtaining an upwardplanar representation (UPR) of the input hypergraph where crossings are modeled by dummy nodes. Furthermore, we present the new problem of computing an orthogonal upward drawing with minimal number of crossings from such an UPR, and show that it can be solved efficiently by providing a simple method.