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15
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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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.
Decremental Biconnectivity on Planar Graphs
"... : In this paper we present a (randomized) algorithm for maintaining the biconnected components of a dynamic planar graph of n vertices under deletions of edges. The biconnected components can be maintained under any sequence of edge deletions in a total of O(n log n) time, with high probability. Thi ..."
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the running time. As in [6], we only need O(n) space. Finally we describe some simply additional operations on the decremental data structure. By aid of them this the data structure is applicable for finding efficiently a \Deltaspanning tree in a biconnected planar graph with a maximum degree 2\Delta \Gamma
Optimal Decremental Connectivity in Planar Graphs∗
"... We show an algorithm for dynamic maintenance of connectivity information in an undirected planar graph subject to edge deletions. Our algorithm may answer connectivity queries of the form ‘Are vertices u and v connected with a path? ’ in constant time. The queries can be intermixed with any sequence ..."
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We show an algorithm for dynamic maintenance of connectivity information in an undirected planar graph subject to edge deletions. Our algorithm may answer connectivity queries of the form ‘Are vertices u and v connected with a path? ’ in constant time. The queries can be intermixed with any
Rolling Upward Planarity Testing of Strongly Connected Graphs
"... A graph is upward planar if it can be drawn without edge crossings such that all edges point upward. Upward planar graphs have been studied on the plane, the standing and rolling cylinders. For all these surfaces, the respective decision problem N Phard in general. Efficient algorithms exist if the ..."
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Cited by 1 (1 self)
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if the graph contains a single source and a single sink, but only for the plane and standing cylinder Here we show that there is a lineartime algorithm to test whether a strongly connected graph is upward planar on the rolling cylinder. For our algorithm, we introduce dual and directed SPQRtrees
Inserting an Edge Into a Planar Graph
 Algorithmica
, 2000
"... Computing a crossing minimum drawing of a given planar graph G augmented by an additional edge e in which all crossings involve e, has been a long standing open problem in graph drawing. Alternatively, the problem can be stated as finding a planar combinatorial embedding of a planar graph G in which ..."
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Cited by 20 (10 self)
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in which the given edge e can be inserted with the minimum number of crossings. Many problems concerned with the optimization over the set of all combinatorial embeddings of a planar graph turned out to be NPhard. Surprisingly, we found a conceptually simple linear time algorithm based on SPQRtrees
Planar Bus Graphs
"... Bus graphs are used for the visualization of hypergraphs, for example in VLSI design. Formally, they are specified by bipartite graphs G = (B ∪ V,E). The bus vertices B are realized by horizontal and vertical segments, and the point vertices V are connected orthogonally to the bus segments without a ..."
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on the computation of maximum matching on some auxiliary graph. Given a good partition we can construct a noncrossing realization of the bus graph on an O(n)×O(n) grid in linear time. In the second part we use SPQRtrees to solve the problem for general planar bipartite graphs.
Algorithm and experiments in testing planar graphs for isomorphism
 Journal of Graph Algorithms and Applications
"... Abstract We give an algorithm for isomorphism testing of planar graphs suitable for practical implementation. The algorithm is based on the decomposition of a graph into biconnected components and further into SPQRtrees. We provide a proof of the algorithm's correctness and a complexity analy ..."
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Cited by 7 (0 self)
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Abstract We give an algorithm for isomorphism testing of planar graphs suitable for practical implementation. The algorithm is based on the decomposition of a graph into biconnected components and further into SPQRtrees. We provide a proof of the algorithm's correctness and a complexity
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 6 (5 self)
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, 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.
246 Abstract Inserting an Edge Into a Planar Graph
"... Computing a crossing minimum drawing of a given planar graph G augmented by an additional edge e in which all crossings involve e, has been a long standing open problem in graph drawing. Alternatively, the problem can be stated as finding a planar combinatorial embedding of a planar graph G in which ..."
Abstract
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in which the given edge e can be inserted with the minimum number of crossings. Many problems concerned with the optimization over the set of all combinatorial embeddings of a planar graph turned out to be NPhard. Surprisingly, we found a conceptually simple linear time algorithm based on SPQRtrees
Testing Mutual Duality of Planar Graphs
, 2013
"... We introduce and study the problem MUTUAL PLANAR DUALITY, which asks for two planar graphs G1 and G2 whether G1 can be embedded such that its dual is isomorphic to G2. Our algorithmic main result is an NPcompleteness proof for the general case and a lineartime algorithm for biconnected graphs. To ..."
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a succinct representation of the equivalence class of a biconnected planar graph. It is similar to SPQRtrees and represents exactly the graphs that are contained in the equivalence class. The testing algorithm then works by testing in linear time whether two such representations are isomorphic. We
Results 1  10
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15