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Planarizing Graphs - A Survey and Annotated Bibliography

Annegret Liebers
2001 Journal of Graph Algorithms and Applications  
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]

Christine Dahn, Nils M. Kriege, Petra Mutzel
2018 Lecture Notes in Computer Science  
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

Carsten Gutwenger, Petra Mutzel, René Weiskircher
2005 Algorithmica  
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]

Martin Gronemann
2016 arXiv   pre-print
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]

Martin Gronemann
2016 Lecture Notes in Computer Science  
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]

Carsten Gutwenger, Michael Jünger, Sebastian Leipert, Petra Mutzel, Merijam Percan, René Weiskircher
2003 Lecture Notes in Computer Science  
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

Hristo N. Djidjev
2006 SIAM Journal on Discrete Mathematics  
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  
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

J. Bruno, K. Steiglitz, L. Weinberg
1970 IEEE Transactions on Circuit Theory  
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]

Giuseppe Battista, Roberto Tamassia, Luca Vismara
1995 Lecture Notes in Computer Science  
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]

Carsten Gutwenger, Petra Mutzel
2004 Lecture Notes in Computer Science  
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]

Christine Dahn, Nils M. Kriege, Petra Mutzel, Julian Schilling
2020 arXiv   pre-print
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

Nabil Hossain, Robert W. McGrail, James Belk, Francesco Matucci
2013 2013 15th International Symposium on Symbolic and Numeric Algorithms for Scientific Computing  
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

Petra Mutzel, René Weiskircher
2006 SIAM Journal on Optimization  
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

F. Liers, G. Pardella
2010 Computational optimization and applications  
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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