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Geometric spanners with small chromatic number

2009
*
Computational geometry
*

We also show that for any > 0, there exists a (1 + )t(k)-

doi:10.1016/j.comgeo.2008.04.003
fatcat:r6msrwefljbtvhnybnitwk6qpa
*spanner*for P that has O(|P |) edges and*chromatic**number*at most k. ... Given an integer k ≥ 2, we consider the problem of computing the smallest real*number*t(k) such that for each set P of points in the plane, there exists a t(k)-*spanner*for P that has*chromatic**number*at ... The*number*of edges in these graphs can be reduced from quadratic to linear*with*a slight increase in the spanning ratio by applying the general technique of Gudmundsson et al. [4] . ...##
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Geometric Spanners With Small Chromatic Number
[article]

2007
*
arXiv
*
pre-print

We also show that for any ϵ >0, there exists a (1+ϵ)t(k)-

arXiv:0711.0114v1
fatcat:ms3nqtgsuzbydexvgannrrucqu
*spanner*for P that has O(|P|) edges and*chromatic**number*at most k. ... Given an integer k ≥ 2, we consider the problem of computing the smallest real*number*t(k) such that for each set P of points in the plane, there exists a t(k)-*spanner*for P that has*chromatic**number*at ... For these reasons, it is desirable to have*geometric*graphs that have both*small**chromatic**number*and*small*stretch factor. ...##
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Geometric Spanners with Small Chromatic Number
[chapter]

*
Approximation and Online Algorithms
*

We also show that for any > 0, there exists a (1 + )t(k)-

doi:10.1007/978-3-540-77918-6_7
dblp:conf/waoa/BoseCCMSZ07
fatcat:r5fllo2cerdj5avdg46uj5zede
*spanner*for P that has O(|P |) edges and*chromatic**number*at most k. ... Given an integer k ≥ 2, we consider the problem of computing the smallest real*number*t(k) such that for each set P of points in the plane, there exists a t(k)-*spanner*for P that has*chromatic**number*at ... The*number*of edges in these graphs can be reduced from quadratic to linear*with*a slight increase in the spanning ratio by applying the general technique of Gudmundsson et al. [4] . ...##
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Robust Geometric Spanners
[article]

2013
*
arXiv
*
pre-print

Informally, when one removes a few vertices from a robust

arXiv:1204.4679v2
fatcat:fl2flb6qy5at3jw3rkt5nkyihu
*spanner*, this harms only a*small**number*of other vertices. ... On the positive side, we give constructions, for any dimension, of robust*spanners**with*a near-linear*number*of edges. ... diameter [8, 9] , planar*spanners*[5, 21, 23, 28] ,*spanners*of low*chromatic**number*[13] , fault-tolerant*spanners*[2, 22, 30, 31] , low-power*spanners*[4, 33, 36] , kinetic*spanners*[1, 3] , angle-constrained ...##
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Robust geometric spanners

2013
*
Proceedings of the 29th annual symposium on Symposuim on computational geometry - SoCG '13
*

Informally, when one removes a few vertices from a robust

doi:10.1145/2493132.2462381
fatcat:2jlv5ebocvgkfbcu5bc43vbvci
*spanner*, this harms only a*small**number*of other vertices. ... On the positive side, we give constructions, for any dimension, of robust*spanners**with*a near-linear*number*of edges. ... diameter [8, 9] , planar*spanners*[5, 21, 23, 29] ,*spanners*of low*chromatic**number*[13] , fault-tolerant*spanners*[2, 22, 31, 32] , lowpower*spanners*[4, 34, 37] , kinetic*spanners*[1, 3] , angle-constrained ...##
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Robust geometric spanners

2013
*
Proceedings of the 29th annual symposium on Symposuim on computational geometry - SoCG '13
*

Informally, when one removes a few vertices from a robust

doi:10.1145/2462356.2462381
dblp:conf/compgeom/BoseDMS13
fatcat:g6jmxnr3q5arrnp7ii4c3b6x54
*spanner*, this harms only a*small**number*of other vertices. ... On the positive side, we give constructions, for any dimension, of robust*spanners**with*a near-linear*number*of edges. ... diameter [8, 9] , planar*spanners*[5, 21, 23, 29] ,*spanners*of low*chromatic**number*[13] , fault-tolerant*spanners*[2, 22, 31, 32] , lowpower*spanners*[4, 34, 37] , kinetic*spanners*[1, 3] , angle-constrained ...##
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Robust Geometric Spanners

2013
*
SIAM journal on computing (Print)
*

Informally, when one removes a few vertices from a robust

doi:10.1137/120874473
fatcat:wg7nxatgenemdkvpuiwjlohtki
*spanner*, this harms only a*small**number*of other vertices. ... On the positive side, we give constructions, for any dimension, of robust*spanners**with*a near-linear*number*of edges. Fig. 1.1. ... diameter [8, 9] , planar*spanners*[5, 21, 23, 29] ,*spanners*of low*chromatic**number*[13] , fault-tolerant*spanners*[2, 22, 31, 32] , lowpower*spanners*[4, 34, 37] , kinetic*spanners*[1, 3] , angle-constrained ...##
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Spanners of Complete k-Partite Geometric Graphs
[article]

2007
*
arXiv
*
pre-print

We address the following problem: Given a complete k-partite

arXiv:0712.0554v1
fatcat:safhlubecjaarcfymjylbendj4
*geometric*graph K whose vertex set is a set of n points in R^d, compute a*spanner*of K that has a "*small*" stretch factor and "few" edges. ... The latter result is optimal: We show that for any 2 ≤ k ≤ n - Θ(√(n n)),*spanners**with*O(n n) edges and stretch factor less than 3 do not exist for all complete k-partite*geometric*graphs. ... Compute a t-*spanner*of the complete k-partite graph K C 1 ...C k that has a "*small*"*number*of edges and whose stretch factor t is "*small*". ...##
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Spanners of Complete k-Partite Geometric Graphs

2009
*
SIAM journal on computing (Print)
*

We address the following problem: Given a complete k-partite

doi:10.1137/070707130
fatcat:d72mofioujhrvnfiqnbu54kj3e
*geometric*graph K whose vertex set is a set of n points in R d , compute a*spanner*of K that has a "*small*" stretch factor and "few" edges. ... The latter result is optimal: We show that there exist complete k-partite*geometric*graphs K such that every subgraph of K*with*a subquadratic*number*of edges has stretch factor at least 3. ... Compute a t-*spanner*of the k-partite complete graph K S 1 ...S k that has a "*small*"*number*of edges and whose stretch factor t is "*small*". ...##
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Constructing Sparse t-Spanners with Small Separators
[chapter]

2003
*
Lecture Notes in Computer Science
*

The

doi:10.1007/978-3-540-45077-1_9
fatcat:enyqrhg7ebbmpdqt7tynklergi
*spanner*also has some additional properties; low weight and a linear*number*of edges. ... Given a set of n points S in the plane and a real value t > 1 we show how to construct in time O(n log n) a t-*spanner*G of S such that there exists a set of vertices S of size O( √ n log n) whose removal ... Thus this graph has a constant*chromatic**number*, and consequently a constant*number*of independent sets. Hence, E i,j is subdivided into a constant*number*of groups, denoted E i,j,k . ...##
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Vertex Fault-Tolerant Geometric Spanners for Weighted Points
[article]

2020
*
arXiv
*
pre-print

in the relative interior of the free space of a polygonal domain P, we detail an algorithm to compute a k-vertex fault-tolerant (4+ϵ)-

arXiv:2011.03354v1
fatcat:bjmwwrc42vcbvaon5rowr7vjy4
*spanner*network*with*O(kn√(h+1)/ϵ^2 n) edges. ... Given a set S of n points, a weight function w to associate a non-negative weight to each point in S, a positive integer k ≥1, and a real*number*ϵ> 0, we present algorithms for computing a*spanner*network ... Apart from the*small**number*of edges,*spanners**with*additional properties such as*small*weight, bounded degree,*small*diameter, planar network, etc., were also considered. ...##
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Spanning Trees and Spanners
[chapter]

2000
*
Handbook of Computational Geometry
*

We survey results in

doi:10.1016/b978-044482537-7/50010-3
fatcat:gitonpgfozgfribivszd6gf5cy
*geometric*network design theory, including algorithms for constructing minimum spanning trees and low-dilation graphs. ... Intuitively, each node of T is then involved only in a*small**number*of pairs, corresponding to nodes*with*similar sized bounding boxes within a similar distance from the node. Lemma 7 (Callahan) . ... For*geometric*graphs in any dimension, they construct*spanners**with*constant dilation and weight O(log n) times that of the minimum spanning tree. ...##
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Local edge colouring of Yao-like subgraphs of Unit Disk Graphs

2009
*
Theoretical Computer Science
*

The

doi:10.1016/j.tcs.2008.11.008
fatcat:v2u4d6fq7fhkdpodooib34mh2e
*number*of colours used is 2kl + 1 and this is optimal for local algorithms (since the maximal degree is 2kl and a colouring*with*2kl colours can only be constructed by a global algorithm), thus showing ... of hops away from it, and hence the algorithm terminates in a constant*number*of steps. ... The minimal such*number*of colors required is called edge*chromatic**number*of the graph. ...##
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Local Construction and Coloring of Spanners of Location Aware Unit Disk Graphs
[chapter]

2008
*
Lecture Notes in Computer Science
*

The output consists of the

doi:10.1007/978-3-540-92248-3_33
fatcat:fotpcq5wqjfdvc4naiqwdprb6e
*spanner*and the 4-coloring. ... Next we look at the colorability of*spanners*of UDGs. In particular we present a local algorithm for constructing a 4-colorable*spanner*of a unit disk graph. ... There are several properties which are desirable for a*spanner*, e.g. connectivity, planarity,*small*node degree,*small*stretch factor and*small*total weight. ...##
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LOCAL CONSTRUCTION AND COLORING OF SPANNERS OF LOCATION AWARE UNIT DISK GRAPHS

2009
*
Discrete Mathematics, Algorithms and Applications (DMAA)
*

The output consists of the

doi:10.1142/s1793830909000415
fatcat:olveez45a5csjoxz4g67igdw6q
*spanner*and the 4-coloring. ... Next we look at the colorability of*spanners*of UDGs. In particular we present a local algorithm for constructing a 4-colorable*spanner*of a unit disk graph. ... There are several properties which are desirable for a*spanner*, e.g. connectivity, planarity,*small*node degree,*small*stretch factor and*small*total weight. ...
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