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Deleting vertices to graphs of bounded genus
[article]

2017
*
arXiv
*
pre-print

We show that a problem

arXiv:1706.04065v1
fatcat:5fdhoa3aejfbnnjrq3ugklvxze
*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 ...##
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Deleting Vertices to Graphs of Bounded Genus

2019
*
Algorithmica
*

We show that a problem

doi:10.1007/s00453-019-00592-7
fatcat:3esnfqs5sncxhh4qod2no325xm
*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. ...##
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A Simpler Linear Time Algorithm for Embedding Graphs into an Arbitrary Surface and the Genus of Graphs of Bounded Tree-Width

2008
*
2008 49th Annual IEEE Symposium on Foundations of Computer Science
*

This algorithm computes the

doi:10.1109/focs.2008.53
dblp:conf/focs/KawarabayashiMR08
fatcat:sqzrlv2ywrge5cxobcfl36uhji
*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. ...##
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Approximation algorithms via contraction decomposition

2010
*
Combinatorica
*

We prove that the edges

doi:10.1007/s00493-010-2341-5
fatcat:2vxvctj6ird7pjltm3emcn7kfm
*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). ...##
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Edge separators for graphs of bounded genus with applications

1993
*
Theoretical Computer Science
*

Many thanks are due also

doi:10.1016/0304-3975(93)90031-n
fatcat:boga3b3zcfh6pltjvrvo5ke77e
*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. ...##
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Planarity Allowing Few Error Vertices in Linear Time

2009
*
2009 50th Annual IEEE Symposium on Foundations of Computer Science
*

Note that deciding the

doi:10.1109/focs.2009.45
dblp:conf/focs/Kawarabayashi09
fatcat:duuzccy2p5dbhicgvlyav7kfry
*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. ...##
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On Algorithms Employing Treewidth for $L$-bounded Cut Problems

2018
*
Journal of Graph Algorithms and Applications
*

Furthermore, we show that the problem remains FPT even for

doi:10.7155/jgaa.00462
fatcat:i5i2ouvbjbfjhesnikggsdibc4
*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*. ...##
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Edge separators for graphs of bounded genus with applications
[chapter]

1992
*
Lecture Notes in Computer Science
*

We prove that every n-vertex

doi:10.1007/3-540-55121-2_15
fatcat:ml3s74yaajbhriteoaheha5f44
*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. ...##
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Linearity of grid minors in treewidth with applications through bidimensionality

2008
*
Combinatorica
*

This strong relationship was previously known for the special cases

doi:10.1007/s00493-008-2140-4
fatcat:cff2tl55u5bubhmz65rewhxpoa
*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. ...##
###
Algorithmic graph embeddings
[chapter]

1995
*
Lecture Notes in Computer Science
*

Based on this new data structure, very efficient algorithms are developed

doi:10.1007/bfb0030829
fatcat:ajeri7qnl5cnreo3ejohdkbiwq
*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. ...##
###
Algorithmic graph embeddings

1997
*
Theoretical Computer Science
*

Based on this new data structure, very efficient algorithms are developed

doi:10.1016/s0304-3975(96)00273-3
fatcat:xi342d6vffgifar277c4wv3534
*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. ...##
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The Asymmetric Traveling Salesman Problem on Graphs with Bounded Genus
[article]

2011
*
arXiv
*
pre-print

We give a constant factor approximation algorithm for the asymmetric traveling salesman problem when the support

arXiv:0909.2849v4
fatcat:dydpol4jcrdongnob5kvukkb6m
*graph**of*the solution*of*the Held-Karp linear programming relaxation has*bounded*orientable ...*genus*. ...*of*this paper. ...##
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Contraction decomposition in h-minor-free graphs and algorithmic applications

2011
*
Proceedings of the 43rd annual ACM symposium on Theory of computing - STOC '11
*

This result is a natural finale

doi:10.1145/1993636.1993696
dblp:conf/stoc/DemaineHK11
fatcat:t47inb45ubbuhjaok46ikjqsqa
*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*...##
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An Algorithmic Meta-Theorem for Graph Modification to Planarity and FOL

2020
*
European Symposium on Algorithms
*

G

doi:10.4230/lipics.esa.2020.51
dblp:conf/esa/FominGST20
fatcat:isihu2it3vgx5ouzbxnjpyyeuq
*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*. ...##
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Pixel and Voxel Representations of Graphs
[article]

2015
*
arXiv
*
pre-print

In 3D, Θ(n^2) voxels are always sufficient and sometimes necessary for any n-vertex

arXiv:1507.01450v1
fatcat:awsjo75qgnct3cr2jqgsmwo4qm
*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. ...
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