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Bibliography of distributed approximation beyond bounded degree
[article]

2021
*
arXiv
*
pre-print

A classic

arXiv:2001.08510v3
fatcat:wjfqufg725fvtluzy62rfjddou
*setting*for such algorithms is*bounded**degree**graphs*, but there is a whole*set*of techniques that have been developed for other classes. ... These classes have a geometric nature (*planar*,*bounded*genus and unit-disk*graphs*) and/or have*bounded*parameters (arboricity, expansion, growth,*independence*) or forbidden structures (forbidden minors ...*Graph*class :*Planar*. 25 A randomized distributed algorithm for the maximal*independent**set*problem*in*growth-*bounded**graphs*- [21] Problem : Maximal*independent**set*. ...##
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Lower bounds for constant degree independent sets

1994
*
Discrete Mathematics
*

Let c(* denote the maximum number of

doi:10.1016/0012-365x(92)00463-2
fatcat:bhb5kda65vhijas2jiz6h356su
*independent*vertices all of which have the same*degree*. We provide lower*bounds*for G(* for*graphs*that are*planar*, maximal*planar*, of*bounded**degree*, or trees. ... Here we consider a restriction: specifically we seek an*independent**set**in*which every vertex has the same*degree*. ... The cardinality of the largest*independent**set*of vertices*in*which all have*degree*j will be denoted by aj = cCj( G). ...##
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Parallel algorithms for fractional and maximal independent sets in planar graphs

1990
*
Discrete Applied Mathematics
*

Our algorithms rely on an efficient parallel algorithm for constructing large

doi:10.1016/0166-218x(90)90130-5
fatcat:sg45zfheqvb4ffz5cwbhrk5p6a
*independent**sets**in**graphs*of*bounded**degree*. ... Using a linear number of EREW processors, the algorithm identifies a maximal*independent**set**in*an arbitrary*planar**graph**in*O(log n log*n) parallel time. ... Acknowledgement This work was supported*in*part by the National Sciences and Engineering Research Council of Canada, grant A3583. ...##
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On Fine-Grained Exact Computation in Regular Graphs
[article]

2021
*
arXiv
*
pre-print

We show that there is no subexponential time algorithm for computing the exact solution of the maximum

arXiv:2008.09008v2
fatcat:efwmqseyqvdmtjff3ljjkke4rq
*independent**set*problem*in*d-regular*graphs*unless ETH fails. ... We utilize the construction to show the NP-hardness of MIS on 5-regular*planar**graphs*, closing the exact complexity status of the problem on regular*planar**graphs*. ... On the other hand, another extension is to*set*up a similar lower*bound**in**planar**graphs*. ...##
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Matchings in 1-planar graphs with large minimum degree
[article]

2020
*
arXiv
*
pre-print

*In*1979, Nishizeki and Baybars showed that every

*planar*

*graph*with minimum

*degree*3 has a matching of size n/3+c (where the constant c depends on the connectivity), and even better

*bounds*hold for

*planar*...

*In*this paper, we investigate similar matching-

*bounds*for 1-

*planar*

*graphs*, i.e.,

*graphs*that can be drawn such that every edge has at most one crossing. ... on the resulting

*independent*

*set*

*in*a 1-

*planar*

*graph*. ...

##
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Large Independent Sets in Triangle-Free Planar Graphs
[article]

2014
*
arXiv
*
pre-print

Every triangle-free

arXiv:1311.2749v2
fatcat:oow4u2wow5cs3cmgf6jf6knm5q
*planar**graph*on n vertices has an*independent**set*of size at least (n+1)/3, and this lower*bound*is tight. ... We give an algorithm that, given a triangle-free*planar**graph*G on n vertices and an integer k>=0, decides whether G has an*independent**set*of size at least (n+k)/3,*in*time 2^O(sqrtk)n. ... Unfortunately, it is unlikely our techniques could be used for*Planar**Independent**Set*-ATLB*in*general*planar**graphs*. ...##
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Large Independent Sets in Triangle-Free Planar Graphs

2017
*
SIAM Journal on Discrete Mathematics
*

Every triangle-free

doi:10.1137/16m1061862
fatcat:gpii6hsasrc4bo73ahis5oxaym
*planar**graph*on n vertices has an*independent**set*of size at least (n + 1)/3, and this lower*bound*is tight. ... We give an algorithm that, given a triangle-free*planar**graph*G on n vertices and an integer k ≥ 0, decides whether G has an*independent**set*of size at least (n + k)/3,*in*time 2 O( √ k) n. ... Unfortunately, it is unlikely our techniques could be used for*Planar**Independent**Set*-ATLB*in*general*planar**graphs*. ...##
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Regular independent sets
[article]

2015
*
arXiv
*
pre-print

The regular

arXiv:1306.5026v3
fatcat:z6s5ysxabjaqhfps4qpf4izira
*independence*number, introduced by Albertson and Boutin*in*1990, is the size of a largest*set*of*independent*vertices with the same*degree*. ... Lower*bounds*were proven for this invariant,*in*terms of the order, for trees and*planar**graphs*. ... An*independent**set*whose vertices all have equal*degree**in*G is called a regular*independent**set*. A k-*independent**set*is a*set*of vertices whose induced subgraph has maximum*degree*at most k. ...##
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Light edges in degree-constrained graphs

2004
*
Discrete Mathematics
*

The method is reÿned

doi:10.1016/j.disc.2003.12.003
fatcat:rf3zwzuhjvbavfx7ndczyifprq
*in*the case of*planar**graphs*to obtain improved*degree**bounds*. ... Let denote the average*degree*, and denote the minimum*degree*of a*graph*. An edge is light if both its endpoints have*degree**bounded*by a constant depending only on and . ... Fig. 2 . 2 The*degree**bound**in*a light matching of · n edges*in*a*planar**graph*. ...##
###
The Power of Orientation in Symmetry-Breaking

2010
*
2010 24th IEEE International Conference on Advanced Information Networking and Applications
*

For instance, we show that

doi:10.1109/aina.2010.98
dblp:conf/aina/PindiproliK10
fatcat:34mcf6gk5fespebzfqy7c6zapa
*in**bounded**degree**graphs*and*planar**graphs*, it is possible to construct a fractional*independent**set*by exchanging O(1) bits. ... Further, we present algorithms to construct maximal*independent**sets**in**bounded**degree**graphs*and oriented trees. ... Towards this end,*in*this work, we consider the problem of finding fractional*independent**sets**in**bounded**degree**graphs*and*planar**graphs*, and, maximal*independent**sets*and k-ruling*sets**in**bounded**degree*...##
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All maximal independent sets and dynamic dominance for sparse graphs

2009
*
ACM Transactions on Algorithms
*

For

doi:10.1145/1597036.1597042
fatcat:pde4phvhbneb7ity7poaofav54
*bounded**degree**graphs*, our algorithms take constant time per*set*generated; for minor-closed*graph*families, the time is O(n) per*set*, and for more general sparse*graph*families we achieve subquadratic ... We can also maintain a dynamic vertex*set**in*an arbitrary m-edge*graph*and test the*independence*of the maintained*set**in*time O(sqrt m) per update. ... Main new results: faster generation for sparse*graphs*O(1) per generated*independent**set*for*bounded**degree**graphs*O(n) per generated*independent**set*for minor-closed*graph*families including*planar**graphs*...##
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Deterministic Algorithms for the Independent Feedback Vertex Set Problem
[chapter]

2015
*
Lecture Notes in Computer Science
*

A feedback vertex

doi:10.1007/978-3-319-19315-1_31
fatcat:vxgx4tfli5cotmq4mk75ujftba
*set*F of an undirected*graph*G is a vertex subset of G whose removal results*in*a forest. Such a*set*F is said to be*independent*if F forms an*independent**set*of G. ... This problem is NP-hard even for*planar*bipartite*graphs*of maximum*degree*four. ... A feedback vertex*set*F of an undirected*graph*G is a vertex subset of G whose removal results*in*a forest. Such a*set*F is said to be*independent*if F forms an*independent**set*of G. ...##
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Parallel construction of subdivision hierarchies

1989
*
Journal of computer and system sciences (Print)
*

The method relies on an efficient parallel algorithm for constructing large

doi:10.1016/0022-0000(89)90042-1
fatcat:zu6hnyrt6rdtril3bn7v2d2rle
*independent**sets**in**planar**graphs*. This is accomplished by a simple reduction to the same problem for lists. ... A direct, simple and general parallel algorithm is described for the preprocessing of a*planar*subdivision for fast (sequential) search. ... ACKNOWLEDGMENT This work was supported*in*part by the National Sciences and Engineering Research Council of Canada Grant A3583. ...##
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Tight Running Time Lower Bounds for Vertex Deletion Problems
[article]

2016
*
arXiv
*
pre-print

We also obtain a dichotomy for running time

arXiv:1511.05449v2
fatcat:5tuy2fyuyjdjhjs7q4qwp6oclq
*bounds*that include the number m of edges*in*the input*graph*: On the one hand, if Π contains all*independent**sets*, then there is no 2^o(n+m)-time algorithm for ... On the other hand, if there is a fixed*independent**set*that is not contained*in*Π and containment*in*Π can determined*in*2^O(n) time or 2^o(m) time, then Π-Vertex Deletion can be solved*in*2^O(√(m))+O( ... time even on*graphs*with*bounded**degree*and thus not on*graphs*with*bounded*degeneracy. ...##
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On the number of simple cycles in planar graphs
[chapter]

1997
*
Lecture Notes in Computer Science
*

Let C (G) denote the number of simple cycles of a

doi:10.1007/bfb0024484
fatcat:24bimg66svbfdkxdb5fsd3j3xy
*graph*G and let C (n) be the maximum of C (G) o ver all*planar**graphs*with n nodes. ... We present a l o wer*bound*on C (n) constructing*graphs*with at least 2:28 n cycles. Applying some probabilistic arguments we p r o ve an upper*bound*of 3:37 n . ... Note that this is only true for triangulated*planar**graphs*. By the four{color{theorem the nodes of G can be partitioned into 4*independent**sets*V 1 ::: V 4 . ...
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