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Planarizing Graphs - A Survey and Annotated Bibliography
2001
Journal of Graph Algorithms and Applications
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While there are many algorithmic results about planarization through edge deletion, the results about vertex splitting, thickness, and crossing number are mostly of a structural nature. ...
Given a finite, undirected, simple graph G, we are concerned with operations on G that transform it into a planar graph. ...
H, S, and K define an instance of Eligible Set Split Planar Graph. G has a planar subgraph with K or more edges if and only if H can be planarized by K vertex splitting operations on S. ...
doi:10.7155/jgaa.00032
fatcat:zftlx3a5jnesxff5hjymjewp5u
A Fixed-Parameter Algorithm for the Max-Cut Problem on Embedded 1-Planar Graphs
[chapter]
2018
Lecture Notes in Computer Science
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Our algorithm computes a maximum cut in an embedded 1-planar graph with n nodes and k edge crossings in time O(3^k · n^3/2 n). ...
Our algorithm recursively reduces a 1-planar graph to at most 3^k planar graphs, using edge removal and node contraction. ...
Split(S2, vyz) 14: else 15: S ← S3 16: end if 17: end if 18: return S Algorithm 1.1: Max-Cut algorithm for embedded 1-planar graphs an arbitrary crossing is selected and removed in three different ways ...
doi:10.1007/978-3-319-94667-2_12
fatcat:4nkl37qqmrbulljobuvvuiiiy4
Inserting an Edge into a Planar Graph
2005
Algorithmica
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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 the given edge e can be inserted with the minimum number of crossings. ...
Algorithm 1 computes an optimal edge insertion path for a biconnected planar graph and two nonadjacent vertices in this graph. ...
doi:10.1007/s00453-004-1128-8
fatcat:pteako5g6zhe3o7k3cfqopvw2a
Bitonic st-orderings for Upward Planar Graphs
[article]
2016
arXiv
pre-print
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We fully characterize planar st-graphs that admit such an ordering and provide a linear-time algorithm for recognition and ordering. ...
If for a graph no bitonic st-ordering exists, we show how to find in linear time a minimum set of edges to split such that the resulting graph admits one. ...
a path Algorithm 2: Algorithm for computing the minimum set of edges to split. input : Embedded planar st-graph G = (V, E) with S(u) for every u ∈ V . output: Minimum set E split ⊂ E to split for admitting ...
arXiv:1608.08578v1
fatcat:lwghubpmhnfzhipvkqebheokpq
Bitonic st-orderings for Upward Planar Graphs
[chapter]
2016
Lecture Notes in Computer Science
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We fully characterize planar st-graphs that admit such an ordering and provide a lineartime algorithm for recognition and ordering. ...
If for a graph no bitonic st-ordering exists, we show how to find in linear time a minimum set of edges to split such that the resulting graph admits one. ...
a path Algorithm 2: Algorithm for computing the minimum set of edges to split. input : Embedded planar st-graph G = (V, E) with S(u) for every u ∈ V . output: Minimum set E split ⊂ E to split for admitting ...
doi:10.1007/978-3-319-50106-2_18
fatcat:cielzn3udfcpli7syegtbvgfli
Subgraph Induced Planar Connectivity Augmentation
[chapter]
2003
Lecture Notes in Computer Science
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The problem arises in automatic graph drawing in the context of c-planarity testing of clustered graphs. ...
Given a planar graph G = (V, E) and a vertex set W ⊆ V , the subgraph induced planar connectivity augmentation problem asks for a minimum cardinality set F of additional edges with end vertices in W such ...
They represent a decomposition of a planar biconnected graph according to its split pairs (pairs of vertices whose removal splits the graph or vertices connected by an edge). ...
doi:10.1007/978-3-540-39890-5_23
fatcat:mrcjncqhh5fw5bztb5bm6vuriu
A Linear-Time Algorithm for Finding a Maximal Planar Subgraph
2006
SIAM Journal on Discrete Mathematics
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We construct an optimal linear-time algorithm for the maximal planar subgraph problem: given a graph G, find a planar subgraph G of G such that adding to G an extra edge of G results in a non-planar graph ...
Our algorithm can be transformed into a new optimal planarity testing algorithm. ...
In Section 3 we develop an algorithm for on-line planarity testing in triconnected graphs in a constant amortized time, which we use as a subroutine in the main algorithm. ...
doi:10.1137/s0895480197328771
fatcat:muvfonqy7nbgdd2hinkjpavvaq
Page 7635 of Mathematical Reviews Vol. , Issue 2000k
[page]
2000
Mathematical Reviews
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In this paper, we prove that, in the case where G is planar and k is an even integer or k = 3, there exists a complete splitting at s such that the resulting graph G’ remains k-edge-connected and planar ...
It is known that all edges incident to s can be split without losing the edge-connectivity of G in V —s. This complete splitting plays an important role in solving many graph connectivity problems. ...
A new planarity test based on 3-connectivity
1970
IEEE Transactions on Circuit Theory
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Initialization of Algorithm Planar Let G be an arbitrary graph. ...
It has been shown by Mac Lane [4] that the graph H is planar if and only if J' and K' are planar graphs. This splitting process is a basic step in Algorithm Planar. ...
doi:10.1109/tct.1970.1083101
fatcat:jsjyquwihraihark5o25zhbbcq
On-line convex planarity testing
[chapter]
1995
Lecture Notes in Computer Science
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An important class of planar straight-line drawings of graphs are the convex drawings, in which all faces are drawn as convex polygons. ...
We consider the problem of testing convex planarity in a semidynamic environment, where a graph is subject to on-line insertions of vertices and edges. ...
Introduction Develop a fully dynamic convex planarity testing algorithm. The best algorithm for fully dynamic planarity testing performs query and update operations in amortized time O( p n) 1 7 ] . ...
doi:10.1007/3-540-59071-4_52
fatcat:foec2fxvqzbj7bqob73oah4mvy
Graph Embedding with Minimum Depth and Maximum External Face
[chapter]
2004
Lecture Notes in Computer Science
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We present new linear time algorithms using the SPQR-tree data structure for computing planar embeddings of planar graphs optimizing certain distance measures. ...
Given a planar graph, the algorithms compute the planar embedding with 1. the minimum depth among the set of all planar embeddings of G, 2. the external face of maximum size among the set of all planar ...
Algorithms for the second step typically deal with planar graphs (i.e., the planarized graphs arising in step 1). ...
doi:10.1007/978-3-540-24595-7_24
fatcat:gjr22axtjncyjjn3phedsxnx7q
Fixed-Parameter Algorithms for the Weighted Max-Cut Problem on Embedded 1-Planar Graphs
[article]
2020
arXiv
pre-print
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Our algorithms compute a maximum cut in an embedded weighted 1-planar graph with n nodes and k edge crossings in time O(3^k · n^3/2log n). ...
Our algorithms recursively reduce a 1-planar graph to at most 3^k planar graphs, using edge removal and node contraction. ...
Weighted Max-Cut algorithm for embedded 1-planar graphs with non-negative edge weigths.
Figure 2 : 2 An example how the algorithm calculates a Max-Cut in an embedded 2-almost-planar graph. ...
arXiv:1812.03074v4
fatcat:zwhccl3ukncehghs6skhhhetgm
Deciding Conjugacy in Thompson's Group F in Linear Time
2013
2013 15th International Symposium on Symbolic and Numeric Algorithms for Scientific Computing
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We prove that our algorithm theoretically achieves an O(n) bound in the size of the input, and we present a O(n 2 ) working solution. ...
This algorithm checks for conjugacy by constructing and comparing directed graphs called strand diagrams. ...
Note that their algorithm accepts planar graphs with loops and multiple edges between vertices. ...
doi:10.1109/synasc.2013.19
dblp:conf/synasc/HossainMBM13
fatcat:42rtnq4n7jbtplyzq4azufdyfq
Bend Minimization in Planar Orthogonal Drawings Using Integer Programming
2006
SIAM Journal on Optimization
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We consider the problem of minimizing the number of bends in a planar orthogonal graph drawing. ...
Our approach for biconnected graphs combines an integer linear programming (ILP) formulation for the set of all embeddings of a planar graph with the network flow formulation for fixed embeddings. ...
Acknowledgment We thank Walter Didimo for providing the code of the branch & bound algorithm and the real world benchmark set. ...
doi:10.1137/040614086
fatcat:q2giaw2mw5gktixad3tczybv7y
Partitioning planar graphs: a fast combinatorial approach for max-cut
2010
Computational optimization and applications
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In this work, we present a new and simple algorithm for determining maximum cuts for arbitrary weighted planar graphs. ...
For planar graphs, there exist several polynomial-time methods determining maximum cuts for arbitrary choice of edge weights. ...
In particular, we thank a referee who has suggested the algorithmic variant for the method of [45] outlined in the end of Section 5. ...
doi:10.1007/s10589-010-9335-5
fatcat:kechjjl52vexhdcrw2pvm76aw4
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