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A Linear Time Algorithm for Finding a Maximal Planar Subgraph Based on PC-Trees [chapter]

Wen-Lian Hsu
2005 Lecture Notes in Computer Science
Our algorithm is based on a simple planarity test [5] developed by the author, which is a vertex addition algorithm based on a depth-first-search ordering.  ...  Given an undirected graph G, the maximal planar subgraph problem is to determine a planar subgraph H of G such that no edge of G-H can be added to H without destroying planarity.  ...  We developed a very simple linear time algorithm [5] based only on a depth-first search tree.  ...

PQ Trees, PC Trees, and Planar Graphs [chapter]

Wen-Lian Hsu, Ross McConnell
2004 Handbook of Data Structures and Applications
The PQ tree play an important role in the linear-time algorithm of Lempel, Even, and Cederbaum for finding a planar embedding of planar graphs [24] .  ...  This step could create a problem when one tries to apply PQ trees to find maximal planar subgraphs of an arbitrary graph [13] .  ...  In addition to giving an algorithm for returning a subdivision of a K 3,3 or a K 5 when the embedding algorithm fails, we have just proven Lemma 1.3, since no planar graph contains such a subdivision.  ...

MultiPlane: A New Framework for Drawing Graphs in Three Dimensions [chapter]

Seok-Hee Hong
2006 Lecture Notes in Computer Science
More specifically, the framework divides a graph into a set of smaller subgraphs, and then draws each subgraph in a 2D plane.  ...  Algorithms that follow this framework vary in computational complexity, depending on the type of graph and the optimisation criteria that are used.  ...  Algorithm for Planar Graphs In this section, we present a linear time algorithm for drawing planar graphs in three dimensions using the new framework.  ...

On the Cutting Edge: Simplified O(n) Planarity by Edge Addition [chapter]

John M. Boyer, Wendy J. Myrvold
2006 Graph Algorithms and Applications 5
We present new O(n)-time methods for planar embedding and Kuratowski subgraph isolation that were inspired by the Booth-Lueker PQ-tree implementation of the Lempel-Even-Cederbaum vertex addition method  ...  In this paper, we improve upon our conference proceedings formulation and upon the Shih-Hsu PC-tree, both of which perform comprehensive tests of planarity conditions embedding the edges from a vertex  ...  The method is based on the PC-tree [23] , a data structure replacement for the PQ-tree which eliminates s, t-numbering, replaces Q-nodes with C-nodes, and detects PC-tree reducibility by testing a number  ...

On the Cutting Edge: Simplified O(n) Planarity by Edge Addition

John M. Boyer, Wendy J. Myrvold
2004 Journal of Graph Algorithms and Applications
We present new O(n)-time methods for planar embedding and Kuratowski subgraph isolation that were inspired by the Booth-Lueker PQ-tree implementation of the Lempel-Even-Cederbaum vertex addition method  ...  In this paper, we improve upon our conference proceedings formulation and upon the Shih-Hsu PC-tree, both of which perform comprehensive tests of planarity conditions embedding the edges from a vertex  ...  The method is based on the PC-tree [23] , a data structure replacement for the PQ-tree which eliminates s, t-numbering, replaces Q-nodes with C-nodes, and detects PC-tree reducibility by testing a number  ...

An Experimental Study of Crossing Minimization Heuristics [chapter]

Carsten Gutwenger, Petra Mutzel
2004 Lecture Notes in Computer Science
We study the effects of various methods for computing a maximal planar subgraph and for edge re-insertion including post-processing and randomization.  ...  The heuristics are based on the planarization approach, so far the most successful framework for crossing minimization.  ...  We thank Sebastian Leipert for providing his implementation of the PQ-based planarity testing, planarity embedding, and planar subgraph algorithms.  ...

Inserting a Vertex into a Planar Graph [chapter]

Markus Chimani, Carsten Gutwenger, Petra Mutzel, Christian Wolf
2009 Proceedings of the Twentieth Annual ACM-SIAM Symposium on Discrete Algorithms
Alternatively, the problem can be stated as finding a planar embedding of G, in which the given star can be inserted requiring the minimum number of crossings.  ...  This algorithm uses the SPQR-tree data structure to handle the exponential number of possible embeddings, in conjunction with dynamic programming schemes for which we introduce partitioning cost subproblems  ...  Preliminaries & Complications SPQR-trees. A block is a maximal 2-connected subgraph of a graph G.  ...

Computing residual connectedness reliability for restricted networks

Charles J. Colbourn, A. Satyanarayana, C. Suffel, K. Sutner
1993 Discrete Applied Mathematics
The classes that admit polynomial time algorithms include trees, series-parallel graphs, partial k-trees, directed path graphs and permutation graphs.  ...  Suppose c is a positive integer and G", is the collection of all graphs G on n points such that every induced subgraph of G on k points has maximum degree at least k-c-l.  ...  It is well known that series-parallel graphs are precisely partial subgraphs of 2-trees; Wald and Colbourn [ 151 gave a simple linear time algorithm for embedding an arbitrary seriesparallel graph in a  ...

Computing and Drawing Isomorphic Subgraphs

Sabine Bachl, Franz J. Brandenburg, Daniel Gmach
2004 Journal of Graph Algorithms and Applications
The isomorphic subgraph problem is finding two disjoint subgraphs of a graph which coincide on at least k edges. The graph is partitioned into a subgraph, its copy, and a remainder.  ...  Then we present two different heuristics for the computation of maximal connected isomorphic subgraphs. Both heuristics use weighting functions and have been tested on four independent test suites.  ...  When restricted to trees the isomorphic subtree problem is solvable in linear time.  ...

Computing and Drawing Isomorphic Subgraphs [chapter]

Sabine Bachl, Franz-Josef Brandenburg
2002 Lecture Notes in Computer Science
The isomorphic subgraph problem is finding two disjoint subgraphs of a graph which coincide on at least k edges. The graph is partitioned into a subgraph, its copy, and a remainder.  ...  Then we present two different heuristics for the computation of maximal connected isomorphic subgraphs. Both heuristics use weighting functions and have been tested on four independent test suites.  ...  When restricted to trees the isomorphic subtree problem is solvable in linear time.  ...

Computing and Drawing Isomorphic Subgraphs [chapter]

S. Bachl, F. -J. Brandenburg, D. Gmach
2006 Graph Algorithms and Applications 5
The isomorphic subgraph problem is finding two disjoint subgraphs of a graph which coincide on at least k edges. The graph is partitioned into a subgraph, its copy, and a remainder.  ...  Then we present two different heuristics for the computation of maximal connected isomorphic subgraphs. Both heuristics use weighting functions and have been tested on four independent test suites.  ...  When restricted to trees the isomorphic subtree problem is solvable in linear time.  ...

Testing Simultaneous Planarity when the Common Graph is 2-Connected [article]

Bernhard Haeupler and Krishnam Raju Jampani and Anna Lubiw
2011 arXiv   pre-print
It is an open problem whether simultaneous planarity can be tested efficiently. We give a linear-time algorithm to test simultaneous planarity when the two graphs share a 2-connected subgraph.  ...  Our algorithm extends to the case of k planar graphs where each vertex [edge] is either common to all graphs or belongs to exactly one of them.  ...  In this paper we give a linear time algorithm to test simultaneous planarity of any two graphs that share a 2-connected subgraph.  ...

An efficient polynomial-time approximation scheme for Steiner forest in planar graphs [chapter]

David Eisenstat, Philip Klein, Claire Mathieu
2012 Proceedings of the Twenty-Third Annual ACM-SIAM Symposium on Discrete Algorithms
We give an O(n log 3 n) approximation scheme for Steiner forest in planar graphs, improving on the previous approximation scheme for this problem, which runs in O(n f ( ) ) time.  ...  , and Marx's algorithm for PC clustering (and our modification of this algorithm), can be implemented in O(n log n) time for planar graphs.  ...  We give a PC-clustering theorem that addresses both issues: the algorithm runs in O(n log n) for planar graphs (in fact, for any excluded-minor family) and it returns subgraphs G 1 , . . . , G with small  ...

An efficient polynomial-time approximation scheme for Steiner forest in planar graphs [article]

David Eisenstat and Philip Klein and Claire Mathieu
2011 arXiv   pre-print
We give an O(n ^3 n) approximation scheme for Steiner forest in planar graphs, improving on the previous approximation scheme for this problem, which runs in O(n^f(ϵ)) time.  ...  , and Marx's algorithm for PC clustering (and our modification of this algorithm), can be implemented in O(n log n) time for planar graphs.  ...  We give a PC-clustering theorem that addresses both issues: the algorithm runs in O(n log n) for planar graphs (in fact, for any excluded-minor family) and it returns subgraphs G 1 , . . . , G with small  ...

Testing Planarity of Partially Embedded Graphs

Patrizio Angelini, Giuseppe Di Battista, Fabrizio Frati, Vít Jelínek, Jan Kratochvíl, Maurizio Patrignani, Ignaz Rutter
2015 ACM Transactions on Algorithms
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  ...  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?  ...  Such a characterization, based on two forbidden topological subgraphs -K 5 and K 3,3 -makes planarity a finite problem and leads to a polynomial time recognition algorithm.  ...
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