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Edge-disjoint spanners in Cartesian products of graphs

2005
*
Discrete Mathematics
*

The parameter c is called the delay

doi:10.1016/j.disc.2005.04.004
fatcat:ydxnp434gbfztb7mgkrd7sn6zi
*of*the*spanner*. We study*edge*-*disjoint**spanners**in**graphs*, focusing on*graphs*formed as*Cartesian**products*. ... Our approach is to construct sets*of**edge*-*disjoint**spanners**in*a*product*based on sets*of**edge*-*disjoint**spanners*and colorings*of*the component*graphs*. ... General*Cartesian**products**In*this section, we present several results on the number*of**edge*-*disjoint**spanners*that can be found*in**graphs*that are the*Cartesian**product**of*other*graphs*. ...##
###
Page 60 of Mathematical Reviews Vol. , Issue 2002A
[page]

2002
*
Mathematical Reviews
*

The pa- rameter c is called the delay

*of*the*spanner*. We investigate the number*of**edge*-*disjoint**spanners**of*a given delay that can exist*in*complete bipartite*graphs*. ... We determine the exact number*of*such*edge*-*disjoint**spanners**of*delay 4 or larger. ...##
###
Self-spanner graphs

2005
*
Discrete Applied Mathematics
*

A subgraph S = (V , E )

doi:10.1016/j.dam.2005.04.004
fatcat:uedgebmttfbw3bssptb5lsu77m
*of*a*graph*G = (V , E) is a k-*spanner**of*G if and only if all*edges*that do not belong to (1) The concept*of**spanners*has been introduced by Peleg and Ullman*in*[24] , where they ... Further results on k-*spanners*and variants thereof can be found for example*in*[18]. k-self-*spanner**In*this section, we examine a class*of**graphs*that guarantees constant delays even*in*the case*of*an ... The next lemma shows that*graphs*that arise from the*Cartesian**product**of*two*graphs*have strong self-*spanner*properties. ...##
###
Sparse Hypercube 3-spanners

2000
*
Discrete Applied Mathematics
*

A t-

doi:10.1016/s0166-218x(99)00246-2
fatcat:wezq7n547vgbvne3aj6xnoanuy
*spanner**of*a*graph*G = (V; E), is a sub-*graph*SG = (V; E ), such that E ⊆ E and for every*edge*{u; v} ∈ E, there is a path from u to v*in*SG*of*length at most t. ... A minimum-*edge*t-*spanner**of*a*graph*G, S G , is the t-*spanner**of*G with the fewest*edges*. ... Acknowledgements The authors gratefully acknowledge the assistance*of*N.C. Wormald for the proof*of*the lower bound*in*Section 4. ...##
###
Contents

2005
*
Discrete Mathematics
*

Stacho

doi:10.1016/s0012-365x(05)00314-6
fatcat:6suufuqjorgfxnwx4djkqep5pu
*Edge*-*disjoint**spanners**in**Cartesian**products**of**graphs*167 Y. Hong and X.-D. Zhang Sharp upper and lower bounds for largest eigenvalue*of*the Laplacian matrices*of*trees 187 A. ... Liu Chromatic sums*of*2-*edge*-connected maps on the plane 211 H. Matsuda On 2-*edge*-connected ½a; b-factors*of**graphs*with Ore-type condition 225 M. Priesler (Moreno) and M. ...##
###
Author index to volume 296

2005
*
Discrete Mathematics
*

Stacho,

doi:10.1016/s0012-365x(05)00315-8
fatcat:m2gwquhb4zcw5h3uc4nmsxjc2u
*Edge*-*disjoint**spanners**in**Cartesian**products**of**graphs*(2-3) 167-186 Golumbic, M.C., H. Kaplan and E. ... Liu, Chromatic sums*of*2-*edge*-connected maps on the plane H., On 2-*edge*-connected ½a; b-factors*of**graphs*with Ore-type condition (2-3) 225-234 Polverino, O., see G. ...##
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Page 2605 of Mathematical Reviews Vol. , Issue 96e
[page]

1996
*
Mathematical Reviews
*

TUEE; Koiice)
The crossing numbers

*of*certain*Cartesian**products*. ... Summary: “A tree ¢-*spanner*, where ¢ is a positive integer,*of*a*graph*G is a spanning tree*in*which the distance between the two ends*of*every*edge**of*G is at most ¢. ...##
###
Spanners for bounded tree-length graphs

2007
*
Theoretical Computer Science
*

This paper concerns construction

doi:10.1016/j.tcs.2007.03.058
fatcat:g3tz2yffcrcthgo3cuekjf2ziu
*of*additive stretched*spanners*with few*edges*for n-vertex*graphs*having a tree-decomposition into bags*of*diameter at most δ, i.e., the tree-length δ*graphs*. ... We also show a lower bound, and prove that there are*graphs**of*tree-length δ for which every multiplicative δ-*spanner*(and thus every additive (δ − 1)-*spanner*) requires Ω (n 1+1/Θ(δ) )*edges*. ... The*Cartesian**product**of*two*graphs*A and B is the*graph*denoted by , x ) , (y, y )) | (x = x and (y, y ) ∈ E(B)) or (y = y and (x, x ) ∈ E(A))}. ...##
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Hop-Spanners for Geometric Intersection Graphs
[article]

2022
*
arXiv
*
pre-print

fat rectangles admits a 2-hop

arXiv:2112.07158v2
fatcat:nxo6kcswingulkeejcmphu5lse
*spanner*with O(nlog n)*edges*, and this bound is the best possible. (3) The intersection*graph**of*n fat convex bodies*in*the plane admits a 3-hop*spanner*with O(nlog n)*edges*... We study t-*spanners*for t∈{2,3} for geometric intersection*graphs**in*the plane. ... We thank Sujoy Bhore for helpful discussions on geometric intersections*graphs*. ...##
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The well-separated pair decomposition for balls
[article]

2017
*
arXiv
*
pre-print

An imprecise t-

arXiv:1706.06287v1
fatcat:dbvhns5orjav3fxpiq3rz5bm54
*spanner*for an imprecise point set R is a*graph*G=(R,E) such that for each precise instance S*of*R,*graph*G_S=(S,E_S), where E_S is the set*of**edges*corresponding to E, is a t-*spanner*. ... Given a real number t>1 and given a set*of*n pairwise*disjoint*d-balls with arbitrary sizes, we use this WSPD to compute*in*O(n n+n/(t-1)^d) time an imprecise t-*spanner*with O(n/(t-1)^d)*edges*for balls ... A d-dimensional hyperrectangle R is the*Cartesian**product**of*d closed intervals. ...##
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Spanners in sparse graphs

2011
*
Journal of computer and system sciences (Print)
*

A t-

doi:10.1016/j.jcss.2010.10.002
fatcat:4sss5iadljgvfbqmf23ge4grme
*spanner**of*a*graph*G is a spanning subgraph S*in*which the distance between every pair*of*vertices is at most t times their distance*in*G. ...*In*particular, for every t 4, the problem*of*finding a tree t-*spanner*is NP-complete on K 6 -minor-free*graphs*. ... The (r, s)-grid is the*Cartesian**product**of*two paths*of*lengths r − 1 and s − 1. ...##
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Lower Bounds on Sparse Spanners, Emulators, and Diameter-reducing shortcuts
[article]

2019
*
arXiv
*
pre-print

By combining these constructions with Abboud and Bodwin's [AbboudB17]

arXiv:1802.06271v2
fatcat:ertgbutfgfdqjbq7tlklzhee2a
*edge*-splitting technique, we get additive stretch lower bounds*of*+Ω(n^1/11) for O(n)-size*spanners*and +Ω(n^1/18) for O(n)-size emulators ... We prove better lower bounds on additive*spanners*and emulators, which are lossy compression schemes for undirected*graphs*, as well as lower bounds on shortcut sets, which reduce the diameter*of*directed ...*In*order to get a lower bound for O(m) shortcuts, we use a*Cartesian**product*combining two such*graphs*layer by layer, forming a sparser*graph*. ...##
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Spanners in Sparse Graphs
[chapter]

2008
*
Lecture Notes in Computer Science
*

A t-

doi:10.1007/978-3-540-70575-8_49
fatcat:ul4bhmcaw5fihek7cgdzshxz7u
*spanner**of*a*graph*G is a spanning subgraph S*in*which the distance between every pair*of*vertices is at most t times their distance*in*G. ...*In*particular, for every t 4, the problem*of*finding a tree t-*spanner*is NP-complete on K 6 -minor-free*graphs*. ... The (r, s)-grid is the*Cartesian**product**of*two paths*of*lengths r − 1 and s − 1. ...##
###
Lower Bounds on Sparse Spanners, Emulators, and Diameter-reducing shortcuts

2018
*
Scandinavian Workshop on Algorithm Theory
*

By combining these constructions with Abboud and Bodwin's [1]

doi:10.4230/lipics.swat.2018.26
dblp:conf/swat/HuangP18
fatcat:5iyk3vkjpfcnxlcoxfkydtadsq
*edge*-splitting technique, we get additive stretch lower bounds*of*+Ω(n 1/13 ) for O(n)-size*spanners*and +Ω(n 1/18 ) for O(n)-size emulators ... We prove better lower bounds on additive*spanners*and emulators, which are lossy compression schemes for undirected*graphs*, as well as lower bounds on shortcut sets, which reduce the diameter*of*directed ...*In*order to get a lower bound for O(m) shortcuts, we use a*Cartesian**product*combining two such*graphs*layer by layer, forming a sparser*graph*. ...##
###
Onk-detour subgraphs of hypercubes

2007
*
Journal of Graph Theory
*

A spanning subgraph G

doi:10.1002/jgt.20281
fatcat:xtckyxoitzfc3lr7qxxipaevya
*of*a*graph*H is a k-detour subgraph*of*H if for each pair*of*vertices x, y ∈ V(H), the distance, dist G ... We view Q n as the*Cartesian**product*Q n 1 × Q n 2 and write every vector v ∈ V (Q n )*in*the form v = (v 1 , v 2 ), where v 1 ∈ V (Q n 1 ) and v 2 ∈ V (Q n 2 ). ... Let H 1 be the subgraph*of*Q n spanned by the*edges*along the coordinates*in*B 0 . Clearly, H 1 is the*disjoint*union*of*2 n−q copies*of*Q q . Step 2. ...
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