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

2005
*
Lecture Notes in Computer Science
*

Our

doi:10.1007/11533719_80
fatcat:nzgstlse75bwvmzbspknee65ni
*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*. ...##
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PQ Trees, PC Trees, and Planar Graphs
[chapter]

2004
*
Handbook of Data Structures and Applications
*

The PQ

doi:10.1201/9781420035179.ch32
fatcat:7osatnyhezghjpkblvomozklw4
*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. ...##
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MultiPlane: A New Framework for Drawing Graphs in Three Dimensions
[chapter]

2006
*
Lecture Notes in Computer Science
*

More specifically, the framework divides

doi:10.1007/11618058_49
fatcat:4753aqzzsvdmpm6k7633ovzjk4
*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. ...##
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On the Cutting Edge: Simplified O(n) Planarity by Edge Addition
[chapter]

2006
*
Graph Algorithms and Applications 5
*

We present new O(n)-

doi:10.1142/9789812773289_0014
fatcat:mywwnckhgvg3fjbed2hlsbkoci
*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 ...##
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On the Cutting Edge: Simplified O(n) Planarity by Edge Addition

2004
*
Journal of Graph Algorithms and Applications
*

We present new O(n)-

doi:10.7155/jgaa.00091
fatcat:vktimhz4kvhativi4mboifsbzi
*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 ...##
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An Experimental Study of Crossing Minimization Heuristics
[chapter]

2004
*
Lecture Notes in Computer Science
*

We study the effects of various methods

doi:10.1007/978-3-540-24595-7_2
fatcat:b4vftqnq3bgj3e3us2mbg2fbue
*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*. ...##
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Inserting a Vertex into a Planar Graph
[chapter]

2009
*
Proceedings of the Twentieth Annual ACM-SIAM Symposium on Discrete Algorithms
*

Alternatively, the problem can be stated as

doi:10.1137/1.9781611973068.42
fatcat:4wzlkbuy3beclhutj4vbwdjv44
*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. ...##
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Computing residual connectedness reliability for restricted networks

1993
*
Discrete Applied Mathematics
*

The classes that admit polynomial

doi:10.1016/0166-218x(93)90233-e
fatcat:6jb4px27mbcxnidv5azyr7s4na
*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*...##
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Computing and Drawing Isomorphic Subgraphs

2004
*
Journal of Graph Algorithms and Applications
*

The isomorphic

doi:10.7155/jgaa.00090
fatcat:vlbqwuwzsbdkhktxhviz4fwgdm
*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]

2002
*
Lecture Notes in Computer Science
*

The isomorphic

doi:10.1007/3-540-36151-0_8
fatcat:saihqp3yqnaevm6qrbjnazxujm
*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]

2006
*
Graph Algorithms and Applications 5
*

The isomorphic

doi:10.1142/9789812773289_0013
fatcat:7deekd4hqraafodsgtbpeuhywu
*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*. ...##
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Testing Simultaneous Planarity when the Common Graph is 2-Connected
[article]

2011
*
arXiv
*
pre-print

It is an open problem whether simultaneous

arXiv:1009.4517v2
fatcat:w2byi7riszdphdpiiazgszsu6u
*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*. ...##
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An efficient polynomial-time approximation scheme for Steiner forest in planar graphs
[chapter]

2012
*
Proceedings of the Twenty-Third Annual ACM-SIAM Symposium on Discrete Algorithms
*

We give an O(n log 3 n) approximation scheme

doi:10.1137/1.9781611973099.53
dblp:conf/soda/EisenstatKM12
fatcat:pfbykxtlsbbk7lqnfkjs2bpc5q
*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 ...##
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An efficient polynomial-time approximation scheme for Steiner forest in planar graphs
[article]

2011
*
arXiv
*
pre-print

We give an O(n ^3 n) approximation scheme

arXiv:1110.1320v2
fatcat:6rhi2flixnchreg5do4vgzmlna
*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 ...##
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Testing Planarity of Partially Embedded Graphs

2015
*
ACM Transactions on Algorithms
*

Our

doi:10.1145/2629341
fatcat:vdhwjcztwzavfe65hilptk6fjy
*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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