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Deleting vertices to graphs of bounded genus [article]

Tomasz Kociumaka, Marcin Pilipczuk
2017 arXiv   pre-print
We show that a problem of deleting a minimum number of vertices from a graph to obtain a graph embeddable on a surface of a given Euler genus is solvable in time 2^C_g · k^2 k n^O(1), where k is the size  ...  of the deletion set, C_g is a constant depending on the Euler genus g of the target surface, and n is the size of the input graph.  ...  We remark that, although our formal statement of Genus Vertex Deletion involves only bounding the Euler genus of the output graph, only minor changes to our algorithms are required if one demands the final  ... 
arXiv:1706.04065v1 fatcat:5fdhoa3aejfbnnjrq3ugklvxze

Deleting Vertices to Graphs of Bounded Genus

Tomasz Kociumaka, Marcin Pilipczuk
2019 Algorithmica  
We show that a problem of deleting a minimum number of vertices from a graph to obtain a graph embeddable on a surface of a given Euler genus is solvable in time 2 C g ·k 2 log k n O(1) , where k is the  ...  size of the deletion set, C g is a constant depending on the Euler genus g of the target surface, and n is the size of the input graph.  ...  , and reproduction in any medium, provided you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons license, and indicate if changes were made.  ... 
doi:10.1007/s00453-019-00592-7 fatcat:3esnfqs5sncxhh4qod2no325xm

A Simpler Linear Time Algorithm for Embedding Graphs into an Arbitrary Surface and the Genus of Graphs of Bounded Tree-Width

Ken-ichi Kawarabayashi, Bojan Mohar, Bruce Reed
2008 2008 49th Annual IEEE Symposium on Foundations of Computer Science  
This algorithm computes the genus and constructs minimum genus embeddings of graphs of bounded tree-width.  ...  Theoretical interest comes from the importance of the genus parameter of graphs and from the fact that graphs of bounded genus naturally generalize the family of planar graphs and share many important  ...  Theorem 1.2 There is a linear time algorithm to compute the genus, the Euler genus, the nonorientable genus and to construct corresponding minimum genus embeddings of graphs of bounded tree-width.  ... 
doi:10.1109/focs.2008.53 dblp:conf/focs/KawarabayashiMR08 fatcat:sqzrlv2ywrge5cxobcfl36uhji

Approximation algorithms via contraction decomposition

Erik D. Demaine, MohammadTaghi Hajiaghayi, Bojan Mohar
2010 Combinatorica  
We prove that the edges of every graph of bounded (Euler) genus can be partitioned into any prescribed number k of pieces such that contracting any piece results in a graph of bounded treewidth (where  ...  We also highlight the only main difficulty in extending our results to general H-minor-free graphs.  ...  The idea is to partition the vertices or edges of the graph into a small number k of pieces such that deleting any one of the pieces results in a bounded-treewidth graph (where the bound depends on k).  ... 
doi:10.1007/s00493-010-2341-5 fatcat:2vxvctj6ird7pjltm3emcn7kfm

Edge separators for graphs of bounded genus with applications

Ondrej Sýkora, Imrich Vrťo
1993 Theoretical Computer Science  
Many thanks are due also to Hristo Djidjev for helpful discussions.  ...  We apply the edge separator to the isoperimetric problem, to efficient embeddings of graphs of genus y into various classes of graphs, including trees, meshes and hypercubes, and to showing lower bounds  ...  The result is a graph Hi of genus < y, / Vi I + 1 vertices and radius s. Apply Lemma 2.1 to the graph Hi.  ... 
doi:10.1016/0304-3975(93)90031-n fatcat:boga3b3zcfh6pltjvrvo5ke77e

Planarity Allowing Few Error Vertices in Linear Time

Ken-ichi Kawarabayashi
2009 2009 50th Annual IEEE Symposium on Foundations of Computer Science  
Note that deciding the genus of k-apex graphs is NP-complete, even for k = 1, as shown by Mohar. Thus k-apex graphs are very different from bounded genus graphs in a sense.  ...  k-apex), and if this is the case, computes a drawing of the graph in the plane after deleting at most k vertices.  ...  We also prove that the deleted graph G i+1 − G (that consists of bounded number of planar graphs of bounded radius in the disks, together with all the vertices in A i+1 − A U ) has bounded tree-width.  ... 
doi:10.1109/focs.2009.45 dblp:conf/focs/Kawarabayashi09 fatcat:duuzccy2p5dbhicgvlyav7kfry

On Algorithms Employing Treewidth for $L$-bounded Cut Problems

Petr Kolman
2018 Journal of Graph Algorithms and Applications  
Furthermore, we show that the problem remains FPT even for bounded genus graphs, a super class of planar graphs.  ...  We show that for planar graphs, it is possible to solve both the edgeand the vertex-deletion version of the problem optimally in O((L+2) 3L n) time.  ...  Theorem 1 was generalized by Eppstein [11] to graphs of bounded genus and this result makes it possible to generalize Theorem 3 also to graphs of bounded genus.  ... 
doi:10.7155/jgaa.00462 fatcat:i5i2ouvbjbfjhesnikggsdibc4

Edge separators for graphs of bounded genus with applications [chapter]

Ondrej Sýkora, Imrich Vrto
1992 Lecture Notes in Computer Science  
We prove that every n-vertex graph of genus 9 and maximal degree k has an edge separator of size O( ..;gTffl). The upper bound is best possible to within a constant factor.  ...  Let G = (V, E) be an n-vertex graph of genus 9 > 0 and maximal degree k whose vertices have nonnegative weights summing to 1 such that no weight exceed 2/3.  ...  The most interesting seems to be the problem of drawing of K n , Km,n and Qn in the surface of orientable genus g.  ... 
doi:10.1007/3-540-55121-2_15 fatcat:ml3s74yaajbhriteoaheha5f44

Linearity of grid minors in treewidth with applications through bidimensionality

Erik D. Demaine, Mohammadtaghi Hajiaghayi
2008 Combinatorica  
This strong relationship was previously known for the special cases of planar graphs and bounded-genus graphs, and is known not to hold for general graphs.  ...  ) [16]. 1 The r×r grid is the planar graph with r 2 vertices arranged on a square grid and with edges connecting horizontally and vertically adjacent vertices.  ...  Seymour for many helpful discussions and for providing a portal into the Graph Minor Theory and revealing some of its hidden structure that we use in this paper.  ... 
doi:10.1007/s00493-008-2140-4 fatcat:cff2tl55u5bubhmz65rewhxpoa

Algorithmic graph embeddings [chapter]

Jianer Chen
1995 Lecture Notes in Computer Science  
Based on this new data structure, very efficient algorithms are developed to solve the problem "given a graph G and an integer k, construct a genus k embedding for the graph G" for a large range of integers  ...  k and for a large class of graphs G  ...  Acknowledgements The author would like to thank Professor Jonathan Gross and Professor Ming Li for their encouragement and comments. He is also grateful to Dr.  ... 
doi:10.1007/bfb0030829 fatcat:ajeri7qnl5cnreo3ejohdkbiwq

Algorithmic graph embeddings

Jianer Chen
1997 Theoretical Computer Science  
Based on this new data structure, very efficient algorithms are developed to solve the problem "given a graph G and an integer k, construct a genus k embedding for the graph G" for a large range of integers  ...  k and for a large class of graphs G  ...  Acknowledgements The author would like to thank Professor Jonathan Gross and Professor Ming Li for their encouragement and comments. He is also grateful to Dr.  ... 
doi:10.1016/s0304-3975(96)00273-3 fatcat:xi342d6vffgifar277c4wv3534

The Asymmetric Traveling Salesman Problem on Graphs with Bounded Genus [article]

Shayan Oveis Gharan, Amin Saberi
2011 arXiv   pre-print
We give a constant factor approximation algorithm for the asymmetric traveling salesman problem when the support graph of the solution of the Held-Karp linear programming relaxation has bounded orientable  ...  genus.  ...  of this paper.  ... 
arXiv:0909.2849v4 fatcat:dydpol4jcrdongnob5kvukkb6m

Contraction decomposition in h-minor-free graphs and algorithmic applications

Erik D. Demaine, MohammadTaghi Hajiaghayi, Ken-ichi Kawarabayashi
2011 Proceedings of the 43rd annual ACM symposium on Theory of computing - STOC '11  
This result is a natural finale to research in contraction decomposition, generalizing previous such decompositions for planar and bounded-genus graphs, and solving the main open problem in this area (  ...  Graph Minor decomposition theorem in the context of edge contractions (without edge deletions).  ...  We then delete the vertices that are both in one of these at most five special child pieces and in the bounded-genus part of Gi+1, from the bounded-genus part of Gi+1 (thus we delete at most 15 vertices  ... 
doi:10.1145/1993636.1993696 dblp:conf/stoc/DemaineHK11 fatcat:t47inb45ubbuhjaok46ikjqsqa

An Algorithmic Meta-Theorem for Graph Modification to Planarity and FOL

Fedor V. Fomin, Petr A. Golovach, Giannos Stamoulis, Dimitrios M. Thilikos, Peter Sanders, Fabrizio Grandoni, Grzegorz Herman
2020 European Symposium on Algorithms  
G to a graph in 𝒫 after applying k times the operation ⊠ on G.  ...  ϕ, the Graph ⊠-Modification to Planarity and ϕ is solvable in f(k,|ϕ|)⋅n² time.  ...  The class of graphs of bounded local treewidth contains graphs of bounded degree, planar graphs, graphs of bounded genus, and apex-minor-free graphs.  ... 
doi:10.4230/lipics.esa.2020.51 dblp:conf/esa/FominGST20 fatcat:isihu2it3vgx5ouzbxnjpyyeuq

Pixel and Voxel Representations of Graphs [article]

Muhammad Jawaherul Alam, Thomas Bläsius, Ignaz Rutter, Torsten Ueckerdt, Alexander Wolff
2015 arXiv   pre-print
In 3D, Θ(n^2) voxels are always sufficient and sometimes necessary for any n-vertex graph. We improve this bound to Θ(n·τ) for graphs of treewidth τ and to O((g+1)^2n^2n) for graphs of genus g.  ...  Then, we bound representation sizes needed for certain graph classes. In 2D, we show that, for k-outerplanar graphs with n vertices, Θ(kn) pixels are always sufficient and sometimes necessary.  ...  We improve this bound to Θ(n · τ) for graphs of treewidth τ and to O((g + 1) 2 n log 2 n) for graphs of genus g.  ... 
arXiv:1507.01450v1 fatcat:awsjo75qgnct3cr2jqgsmwo4qm
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