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Bibliography of distributed approximation beyond bounded degree
[article]
2021
arXiv
pre-print
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A classic 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. ...
arXiv:2001.08510v3
fatcat:wjfqufg725fvtluzy62rfjddou
Lower bounds for constant degree independent sets
1994
Discrete Mathematics
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Let c(* denote the maximum number of 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). ...
doi:10.1016/0012-365x(92)00463-2
fatcat:bhb5kda65vhijas2jiz6h356su
Parallel algorithms for fractional and maximal independent sets in planar graphs
1990
Discrete Applied Mathematics
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Our algorithms rely on an efficient parallel algorithm for constructing large 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. ...
doi:10.1016/0166-218x(90)90130-5
fatcat:sg45zfheqvb4ffz5cwbhrk5p6a
On Fine-Grained Exact Computation in Regular Graphs
[article]
2021
arXiv
pre-print
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2020 pre-print arXiv:2008.09008v1 fatcat:enes2dompfajpolfotvs7yx6zy
We show that there is no subexponential time algorithm for computing the exact solution of the maximum 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. ...
arXiv:2008.09008v2
fatcat:efwmqseyqvdmtjff3ljjkke4rq
Matchings in 1-planar graphs with large minimum degree
[article]
2020
arXiv
pre-print
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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. ...
arXiv:1911.04603v2
fatcat:xygcglzw7nf37grwcrg7kq3imy
Large Independent Sets in Triangle-Free Planar Graphs
[article]
2014
arXiv
pre-print
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Every triangle-free 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. ...
arXiv:1311.2749v2
fatcat:oow4u2wow5cs3cmgf6jf6knm5q
Large Independent Sets in Triangle-Free Planar Graphs
2017
SIAM Journal on Discrete Mathematics
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Every triangle-free 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. ...
doi:10.1137/16m1061862
fatcat:gpii6hsasrc4bo73ahis5oxaym
Regular independent sets
[article]
2015
arXiv
pre-print
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The regular 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. ...
arXiv:1306.5026v3
fatcat:z6s5ysxabjaqhfps4qpf4izira
Light edges in degree-constrained graphs
2004
Discrete Mathematics
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The method is reÿned 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. ...
doi:10.1016/j.disc.2003.12.003
fatcat:rf3zwzuhjvbavfx7ndczyifprq
The Power of Orientation in Symmetry-Breaking
2010
2010 24th IEEE International Conference on Advanced Information Networking and Applications
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For instance, we show that 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 ...
doi:10.1109/aina.2010.98
dblp:conf/aina/PindiproliK10
fatcat:34mcf6gk5fespebzfqy7c6zapa
All maximal independent sets and dynamic dominance for sparse graphs
2009
ACM Transactions on Algorithms
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For 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 ...
doi:10.1145/1597036.1597042
fatcat:pde4phvhbneb7ity7poaofav54
Deterministic Algorithms for the Independent Feedback Vertex Set Problem
[chapter]
2015
Lecture Notes in Computer Science
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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. ...
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. ...
doi:10.1007/978-3-319-19315-1_31
fatcat:vxgx4tfli5cotmq4mk75ujftba
Parallel construction of subdivision hierarchies
1989
Journal of computer and system sciences (Print)
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The method relies on an efficient parallel algorithm for constructing large 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. ...
doi:10.1016/0022-0000(89)90042-1
fatcat:zu6hnyrt6rdtril3bn7v2d2rle
Tight Running Time Lower Bounds for Vertex Deletion Problems
[article]
2016
arXiv
pre-print
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We also obtain a dichotomy for running time 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. ...
arXiv:1511.05449v2
fatcat:5tuy2fyuyjdjhjs7q4qwp6oclq
On the number of simple cycles in planar graphs
[chapter]
1997
Lecture Notes in Computer Science
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Let C (G) denote the number of simple cycles of a 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 . ...
doi:10.1007/bfb0024484
fatcat:24bimg66svbfdkxdb5fsd3j3xy
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