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Dynamic Planar Embeddings of Dynamic Graphs [article]

Jacob Holm, Eva Rotenberg
2017 arXiv   pre-print
We present an algorithm to support the dynamic embedding in the plane of a dynamic graph.  ...  Our time bounds match those of Italiano et al. for a static (flipless) embedding of a dynamic graph.  ...  Introduction We present a data structure for supporting and maintaining a dynamic planar embedding of a dynamic graph.  ... 
arXiv:1704.00565v1 fatcat:6wz7is7eyfedlpybdxnjuqrhvy

Dynamic Planar Embeddings of Dynamic Graphs

Jacob Holm, Eva Rotenberg
2017 Theory of Computing Systems  
We present an algorithm to support the dynamic embedding in the plane of a dynamic graph.  ...  Our time bounds match those of Italiano et al. for a static (flipless) embedding of a dynamic graph.  ...  Introduction We present a data structure for supporting and maintaining a dynamic planar embedding of a dynamic graph.  ... 
doi:10.1007/s00224-017-9768-7 fatcat:mz4vziny5neg5ndy4nipw4h24a

Lower Bounds for Fully Dynamic Connectivity Problems in Graphs

M. R. Henzinger, M. L. Fredman
1998 Algorithmica  
We also show an amortized lower bound of (log n/(log log n + log b)) per operation for fully dynamic planarity testing in embedded graphs.  ...  We prove lower bounds on the complexity of maintaining fully dynamic k-edge or k-vertex connectivity in plane graphs and in (k − 1)-vertex connected graphs.  ...  We call a graph G plane if G is planar and we are given a fixed embedding of G. An embedding of a graph G is uniquely determined by fixing at each vertex the order of its incident edges.  ... 
doi:10.1007/pl00009228 fatcat:fuoyf6xrafflnmzyncqunizhxy

Dynamic coloring and list dynamic coloring of planar graphs

Seog-Jin Kim, Sang June Lee, Won-Jin Park
2013 Discrete Applied Mathematics  
Based on Thomassen's result [12] that every planar graph is 5-choosable, an interesting question is whether the list dynamic chromatic number of every planar graph is at most 5 or not.  ...  A dynamic chromatic number χ d (G) of a graph G is the least number k such that G has a proper k-coloring of the vertex set V (G) so that for each vertex of degree at least 2, its neighbors receive at  ...  Let G be a 2-connected planar graph other than odd cycles. Fix a planar embedding of G, and for simplicity, denote the embedding by G. (Now G is a plane graph.)  ... 
doi:10.1016/j.dam.2013.03.005 fatcat:2s3ynd6vv5gdpaj3te5xdkfi3u

Fast Minor Testing in Planar Graphs

Isolde Adler, Frederic Dorn, Fedor V. Fomin, Ignasi Sau, Dimitrios M. Thilikos
2011 Algorithmica  
Our technique is based on a novel approach called partially embedded dynamic programming.  ...  We give an algorithm that, given a planar n-vertex graph G and an h-vertex graph H, either finds in time 2 O(h) · n + O(n 2 · log n) a model of H in G, or correctly concludes that G does not contain H  ...  In order to prove Theorem 1, we introduce a novel approach of dynamic programming in planar graphs of bounded branchwidth, namely partially embedded dynamic programming.  ... 
doi:10.1007/s00453-011-9563-9 fatcat:dwq3ombtkjalpegm5fs7rly6cu

Algorithms for the Minimum Edge Cover of H-Subgraphs of a Graph [chapter]

Alexander Grigoriev, Bert Marchal, Natalya Usotskaya
2010 Lecture Notes in Computer Science  
Using the constructed dynamic programs, we design a Baker's type approximation scheme for the problem on planar graphs.  ...  First, we derive a polynomial time dynamic program for the problem on graphs of bounded treewidth and bounded maximum vertex degree.  ...  Such a drawing is called a planar embedding of the graph. A planar embedding in which all vertices are on the exterior face is called a 1-outerplanar embedding.  ... 
doi:10.1007/978-3-642-11266-9_38 fatcat:ydr4lcgscjg7xbdnebe5ufn5tq

Editorial: Special Issue on "Theoretical Aspects of Computer Science" (STACS 2015)

Ernst W. Mayr
2017 Theory of Computing Systems  
The paper Dynamic Planar Embeddings of Dynamic Graphs by Rotenberg and Holm addresses a similar problem area, concerning dynamic embeddings of (planar) graphs. where the authors present significant simplifications  ...  Jakub Łacki and Piotr Sankowski: Optimal Decremental Connectivity in Planar Graphs 9. Eva Rotenberg and Jacob Holm: Dynamic Planar Embeddings of Dynamic Graphs 10.  ... 
doi:10.1007/s00224-017-9793-6 fatcat:vwlg737kdngn7imwdsac2oiyum

A Fully Dynamic Algorithm to Test the Upward Planarity of Single-Source Embedded Digraphs [chapter]

Aimal Rextin, Patrick Healy
2009 Lecture Notes in Computer Science  
In this paper, we present a dynamic algorithm that checks if a singlesource embedded digraph is upward planar in the presence of edge insertions and edge deletions.  ...  Let G φ be an upward planar single-source embedded digraph and let G φ be a single-source embedded digraph obtained by updating G φ .  ...  Dynamic graph algorithms are useful when a graph has discrete changes like the addition or deletion of vertices or edges.  ... 
doi:10.1007/978-3-642-00219-9_24 fatcat:vwi53nhfqvcr5jrzjgze6mzv7e

Popular Conjectures as a Barrier for Dynamic Planar Graph Algorithms [article]

Amir Abboud, Søren Dahlgaard
2016 arXiv   pre-print
The dynamic shortest paths problem on planar graphs asks us to preprocess a planar graph G such that we may support insertions and deletions of edges in G as well as distance queries between any two nodes  ...  planar graphs.  ...  This holds even if the planar embedding of G never changes. Finally, we use our framework to show lower bounds for various other problems, like dynamic girth and diameter.  ... 
arXiv:1605.03797v1 fatcat:nvi2uvrd25dbrhwg6wgz5x6tsu

Planar Subgraph Isomorphism Revisited [article]

Frederic Dorn
2009 arXiv   pre-print
We introduce the technique of "embedded dynamic programming" on a suitably structured graph decomposition, which exploits the topology of the underlying embeddings of the subgraph pattern (rather than  ...  of the host graph).  ...  The author thanks Paul Bonsma, Holger Dell and Fedor Fomin for discussions and comments of great value to the presentation of these results.  ... 
arXiv:0909.4692v1 fatcat:ceulzvtlm5a2jdbw3enldxn3hi

Fast Minor Testing in Planar Graphs [chapter]

Isolde Adler, Frederic Dorn, Fedor V. Fomin, Ignasi Sau, Dimitrios M. Thilikos
2010 Lecture Notes in Computer Science  
Our technique is based on a novel approach called partially embedded dynamic programming. An extended abstract of this work appeared in the proceedings of ESA'10 [2].  ...  We give an algorithm that, given a planar n-vertex graph G and an h-vertex graph H, either finds in time O(2 O(h) · n + n 2 · log n) a model of H in G, or correctly concludes that G does not contain H  ...  In order to prove Theorem 1, we introduce a novel approach of dynamic programming in planar graphs of bounded branchwidth, namely partially embedded dynamic programming.  ... 
doi:10.1007/978-3-642-15775-2_9 fatcat:fyf6lxyrdvcj5frvn44uixkix4

Page 5576 of Mathematical Reviews Vol. , Issue 94j [page]

1994 Mathematical Reviews  
This result is applied to the study of embeddings of planar graphs into general surfaces. {For the entire collection see MR 93m:05005.} V. R.  ...  Summary: “We present dynamic algorithms for embedding com- plete and arbitrary dynamic d-ary trees into the de Bruijn graph DG(d,k) of base d and order k, a very attractive interconnection topology for  ... 

Sketch-Driven Orthogonal Graph Drawing [chapter]

Ulrik Brandes, Markus Eiglsperger, Michael Kaufmann, Dorothea Wagner
2002 Lecture Notes in Computer Science  
We present an orthogonal graph drawing algorithm that uses a sketchy drawing of the graph as input.  ...  Potential applications for this kind of drawing algorithm include the generation of schematic maps from geographic networks and interactive orthogonal graph drawing.  ...  drawings of embedded planar graphs with maximum degree four [23] .  ... 
doi:10.1007/3-540-36151-0_1 fatcat:upwbdwh2tjgw3pblet7ysemxbm

Layer-free upward crossing minimization

Markus Chimani, Carsten Gutwenger, Petra Mutzel, Hoi-Ming Wong
2010 ACM Journal of Experimental Algorithmics  
An upward drawing of a DAG G is a drawing of G in which all edges are drawn as curves increasing monotonically in the vertical direction.  ...  Our algorithm is based on a two-stage upward planarization approach, which computes a feasible upward planar subgraph in the first step, and re-inserts the remaining edges by computing constraint-feasible  ...  Let Γ 1 be an upward planar embedding of U 1 . We say Γ 1 induces an upward planar embedding Γ of U , which is obtained by reversing the realization procedure while maintaining the embedding.  ... 
doi:10.1145/1671970.1671975 fatcat:ehm7vp4oozd6zph76b2engjuie

Decremental SPQR-trees for Planar Graphs

Jacob Holm, Giuseppe F. Italiano, Adam Karczmarz, Jakub Lacki, Eva Rotenberg, Michael Wagner
2018 European Symposium on Algorithms  
We present a decremental data structure for maintaining the SPQR-tree of a planar graph subject to edge contractions and deletions. The update time, amortized over Ω(n) operations, is O(log 2 n).  ...  Via SPQR-trees, we give a decremental data structure for maintaining 3-vertex connectivity in planar graphs.  ...  Let G be a plane graph (a planar graph embedded in the plane).  ... 
doi:10.4230/lipics.esa.2018.46 dblp:conf/esa/HolmIKLR18 fatcat:styzxed7dffvzlf5lp4zem25uu
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