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Collective tree spanners and routing in ATfree related graphs (Extended Abstract)
 IN GRAPHTHEORETIC CONCEPTS IN COMPUTER SCIENCE, LECTURE NOTES IN COMPUT. SCI. 3353
, 2004
"... In this paper we study collective additive tree spanners for families of graphs that either contain or are contained in ATfree graphs. We say that a graph G = (V, E) admits a system of µ collective additive tree rspanners if there is a system T (G) of at most µ spanning trees of G such that for any ..."
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Cited by 9 (8 self)
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In this paper we study collective additive tree spanners for families of graphs that either contain or are contained in ATfree graphs. We say that a graph G = (V, E) admits a system of µ collective additive tree rspanners if there is a system T (G) of at most µ spanning trees of G such that for any two vertices x, y of G a spanning tree T ∈ T (G) exists such that dT(x, y) ≤ dG(x, y) + r. Among other results, we show that ATfree graphs have a system of two collective additive tree 2spanners (whereas there are trapezoid graphs that do not admit any additive tree 2spanner). Furthermore, based on this collection, we derive a compact and efficient routing scheme. Also, any DSPgraph (there exists a dominating shortest path) admits an additive tree 4spanner, a system of two collective additive tree 3spanners and a system of five collective additive tree 2spanners.
Collective Tree Spanners in Graphs with Bounded Parameters
 ALGORITHMICA
, 2006
"... In this paper we study collective additive tree spanners for special families of graphs including planar graphs, graphs with bounded genus, graphs with bounded treewidth, graphs with bounded cliquewidth, and graphs with bounded chordality. We say that a graph G = (V, E) admits a system of μ colle ..."
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Cited by 4 (4 self)
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In this paper we study collective additive tree spanners for special families of graphs including planar graphs, graphs with bounded genus, graphs with bounded treewidth, graphs with bounded cliquewidth, and graphs with bounded chordality. We say that a graph G = (V, E) admits a system of μ collective additive tree rspanners if there is a system T (G) of at most μ spanning trees of G such that for any two vertices x,y of G a spanning tree T ∈ T (G) exists such that dT (x, y) ≤ dG(x, y) + r. We describe a general method for constructing a “small” system of collective additive tree rspanners with small values of r for “well ” decomposable graphs, and as a byproduct show (among other results) that any weighted planar graph admits a system of O ( √ n) collective additive tree 0spanners, any weighted graph with treewidth at most k − 1 admits a system of k log 2 n collective additive tree 0spanners, any weighted graph with cliquewidth at most k admits a system of k log 3/2 n collective additive tree (2w)spanners, and any weighted graph with size of largest induced cycle at most c admits a system of log 2 n collective additive tree (2⌊c/2⌋w)spanners and a system of 4 log 2 n collective additive tree (2(⌊c/3⌋+1)w)spanners (here, w is the maximum edge weight in G). The latter result is refined for weighted weakly chordal graphs: any such graph admits a system of 4 log 2 n collective additive tree (2w)spanners. Furthermore, based on this collection of trees, we derive a compact and efficient routing scheme for those families of graphs. Results of this paper were partially presented at the ISAAC’05 conference [14].
Spanners for bounded treelength graphs
, 2007
"... This paper concerns construction of additive stretched spanners with few edges for nvertex graphs having a treedecomposition into bags of diameter at most δ, i.e., the treelength δ graphs. For such graphs we construct additive 2δspanners with O(δn+n log n) edges, and additive 4δspanners with O( ..."
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Cited by 3 (2 self)
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This paper concerns construction of additive stretched spanners with few edges for nvertex graphs having a treedecomposition into bags of diameter at most δ, i.e., the treelength δ graphs. For such graphs we construct additive 2δspanners with O(δn+n log n) edges, and additive 4δspanners with O(δn) edges. This provides new upper bounds for chordal graphs for which δ = 1. We also show a lower bound, and prove that there are graphs of treelength δ for which every multiplicative δspanner (and thus every additive (δ − 1)spanner) requires Ω(n 1+1/Θ(δ) ) edges.
On kDetour Subgraphs of Hypercubes
, 2007
"... A spanning subgraph G of a graph H is a kdetour subgraph of H if for each pair of vertices x, y ∈ V(H), the distance, distG(x, y), between x and y in G exceeds that in H by at most k. Such subgraphs sometimes also are called additive spanners. In this article, we study kdetour subgraphs of the n ..."
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Cited by 2 (0 self)
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A spanning subgraph G of a graph H is a kdetour subgraph of H if for each pair of vertices x, y ∈ V(H), the distance, distG(x, y), between x and y in G exceeds that in H by at most k. Such subgraphs sometimes also are called additive spanners. In this article, we study kdetour subgraphs of the ndimensional cube, Q n, with few edges or with moderate maximum degree. Let �(k, n) denote the minimum possible maximum degree of a kdetour subgraph of Q n. The main result is that for every k ≥ 2 and
Navigating in a graph by aid of its spanning tree
, 2008
"... Let G =(V,E) be a graph and T be a spanning tree of G. We consider the following strategy in advancing in G from a vertex x towards a target vertex y: from a current vertex z (initially, z = x), unless z = y, gotoaneighborofz in G that is closest to y in T (breaking ties arbitrarily). In this stra ..."
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Cited by 1 (1 self)
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Let G =(V,E) be a graph and T be a spanning tree of G. We consider the following strategy in advancing in G from a vertex x towards a target vertex y: from a current vertex z (initially, z = x), unless z = y, gotoaneighborofz in G that is closest to y in T (breaking ties arbitrarily). In this strategy, each vertex has full knowledge of its neighborhood in G and can use the distances in T to navigate in G. Thus, additionally to standard local information (the neighborhood NG(v)), the only global information that is available to each vertex v is the topology of the spanning tree T (in fact, v can know only a very small piece of information about T and still be able to infer from it the necessary treedistances). For each source vertex x and target vertex y, this way, a path, called a greedy routing path, is produced. Denote by gG,T (x, y) the length of a longest greedy routing path that can be produced for x and y using this strategy and T. We say that a spanning tree T of a graph G is an additive rcarcass for G if gG,T (x, y) ≤ dG(x, y)+r for each ordered pair x, y ∈ V. In this paper, we investigate the problem, given a graph family F, whether a small integer r exists such that any graph G ∈Fadmits an additive rcarcass. We show that rectilinear p × q grids, hypercubes, distancehereditary graphs, dually chordal graphs (and, therefore, strongly chordal graphs and interval graphs), all admit additive 0carcasses. Furthermore, every chordal graph G admits an additive (ω(G) + 1)carcass (where ω(G) is the size of a maximum clique of G), each 3sunfree chordal graph admits an additive 2carcass, each chordal bipartite graph admits an additive 4carcass. In particular, any ktree admits an additive (k+2)carcass. All those carcasses are easy to construct.
A New Optimality Measure for Distance Dominating Sets
"... I hereby declare that I am the sole author of this thesis. This is a true copy of the thesis, including any required final revisions, as accepted by my examiners. I understand that my thesis may be made electronically available to the public. ii We study the problem of finding the smallest power of ..."
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Cited by 1 (0 self)
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I hereby declare that I am the sole author of this thesis. This is a true copy of the thesis, including any required final revisions, as accepted by my examiners. I understand that my thesis may be made electronically available to the public. ii We study the problem of finding the smallest power of an input graph that has k disjoint dominating sets, where the ith power of an input graph G is constructed by adding edges between pairs of vertices in G at distance i or less, and a subset of vertices in a graph G is a dominating set if and only if every vertex in G is adjacent to a vertex in this subset. The problem is a different view of the ddomatic number problem in which the goal is to find the maximum number of disjoint dominating sets in the dth power of the input graph. This problem is motivated by applications in multifacility location and distributed networks. In the facility location framework, for instance, there are k types of services that all clients in different regions of a city should receive. A
Graphs and Combinatorics © SpringerVerlag 2008 On 2Detour Subgraphs of the Hypercube
"... Abstract. A spanning subgraph H of a graph G is a 2detour subgraph of G if for each x, y ∈ V (G), dH (x, y) ≤ dG(x, y) + 2. We prove a conjecture of Erdős, Hamburger, Pippert, and Weakley by showing that for some positive constant c and every n, each2detour subgraph of the ndimensional hypercube ..."
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Abstract. A spanning subgraph H of a graph G is a 2detour subgraph of G if for each x, y ∈ V (G), dH (x, y) ≤ dG(x, y) + 2. We prove a conjecture of Erdős, Hamburger, Pippert, and Weakley by showing that for some positive constant c and every n, each2detour subgraph of the ndimensional hypercube Qn has at least c log 2 n · 2 n edges.
On 2detour subgraphs of the hypercube
"... Abstract. A spanning subgraph H of a graph G is a 2detour subgraph of G if for each x,y ∈ V (G), dH(x,y) ≤ dG(x,y) + 2. We prove a conjecture of Erdős, Hamburger, Pippert, and Weakley by showing that for some positive constant c and every n, each 2detour subgraph of the ndimensional hypercube Qn ..."
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Abstract. A spanning subgraph H of a graph G is a 2detour subgraph of G if for each x,y ∈ V (G), dH(x,y) ≤ dG(x,y) + 2. We prove a conjecture of Erdős, Hamburger, Pippert, and Weakley by showing that for some positive constant c and every n, each 2detour subgraph of the ndimensional hypercube Qn has at least clog 2 n · 2 n edges.
Heuristics for Generating Additive Spanners
, 2004
"... Given an undirected and unweighted graph G, the subgraph S is an additive spanner of G with delay d if the distance between any two vertices in S is no more than d greater than their distance in G. It is known that the problem of finding additive spanners of arbitrary graphs for any fixed value of d ..."
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Given an undirected and unweighted graph G, the subgraph S is an additive spanner of G with delay d if the distance between any two vertices in S is no more than d greater than their distance in G. It is known that the problem of finding additive spanners of arbitrary graphs for any fixed value of d with a minimum number of edges is NPhard. Additive spanners are used as substructures for communication networks which are subject to design constraints such as minimizing the number of connections in the network, or permitting only a maximum number of connections at any one node. In this thesis, we consider the problem of constructing good additive spanners. We say that a spanner is “good ” if it contains few edges, but not necessarily a minimum number of them. We present several algorithms which, given a graph G and a delay parameter d as input, produce a graph S which is an additive spanner of G with delay d. We evaluate each of these algorithms experimentally over a large set of input