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### Edge-disjoint spanners in Cartesian products of graphs

Guillaume Fertin, Arthur L. Liestman, Thomas C. Shermer, Ladislav Stacho
2005 Discrete Mathematics
The parameter c is called the delay 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

Serafino Cicerone, Gabriele Di Stefano, Dagmar Handke
2005 Discrete Applied Mathematics
A subgraph S = (V , E ) 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  , where they  ...  Further results on k-spanners and variants thereof can be found for example in . 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

W. Duckworth, M. Zito
2000 Discrete Applied Mathematics
A t-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 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, 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.  ...

### 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

Yon Dourisboure, Feodor F. Dragan, Cyril Gavoille, Chenyu Yan
2007 Theoretical Computer Science
This paper concerns construction 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))}.  ...

### Hop-Spanners for Geometric Intersection Graphs [article]

Jonathan B. Conroy, Csaba D. Tóth
2022 arXiv   pre-print
fat rectangles admits a 2-hop 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.  ...

### The well-separated pair decomposition for balls [article]

2017 arXiv   pre-print
An imprecise t-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.  ...

### Spanners in sparse graphs

Feodor F. Dragan, Fedor V. Fomin, Petr A. Golovach
2011 Journal of computer and system sciences (Print)
A t-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 [article]

Shang-En Huang, Seth Pettie
2019 arXiv   pre-print
By combining these constructions with Abboud and Bodwin's [AbboudB17] 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.  ...

### Spanners in Sparse Graphs [chapter]

Feodor F. Dragan, Fedor V. Fomin, Petr A. Golovach
2008 Lecture Notes in Computer Science
A t-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

Shang-En Huang, Seth Pettie, Marc Herbstritt
2018 Scandinavian Workshop on Algorithm Theory
By combining these constructions with Abboud and Bodwin's  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

Nana Arizumi, Peter Hamburger, Alexandr Kostochka
2007 Journal of Graph Theory
A spanning subgraph G 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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