In this paper we study collective additive tree spanners for families of graphs that either contain or are contained in AT-free 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 AT-free graphs have a system of two collective additive tree 2-spanners (whereas there are trapezoid graphs that do not admit any additive tree 2-spanner). Furthermore, based on this collection, we derive a compact and efficient routing scheme. Also, any DSP-graph (there exists a dominating shortest path) admits an additive tree 4-spanner, a system of two collective additive tree 3-spanners and a system of five collective additive tree 2-spanners.

In this paper we study collective additive tree spanners for special families of graphs including planar graphs, graphs with bounded genus, graphs with bounded tree-width, graphs with bounded clique-width, and graphs with bounded chordality. We say that a graph G = (V, E) admits a system of μ collective additive tree r-spanners 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 r-spanners 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 0-spanners, any weighted graph with tree-width at most k − 1 admits a system of k log 2 n collective additive tree 0-spanners, any weighted graph with clique-width 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].

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Abstract. 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 tree-distances). 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 r-carcass 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 r-carcass. We show that rectilinear p × q grids, hypercubes, distance-hereditary graphs, dually chordal graphs (and, therefore, strongly chordal graphs and interval graphs), all admit additive 0-carcasses. 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 3-sun-free chordal graph admits an additive 2-carcass, each chordal bipartite graph admits an additive 4-carcass. In particular, any k-tree admits an additive (k+2)-carcass. All those carcasses are easy to construct. 1

Abstract. A spanning subgraph H of a graph G is a 2-detour 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 2-detour subgraph of the n-dimensional hypercube Qn has at least clog 2 n · 2 n edges.

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 NP-hard. 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

Abstract: 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, 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 k-detour subgraphs of the n-dimensional cube, Q n, with few edges or with moderate maximum degree. Let �(k, n) denote the minimum possible maximum degree of a k-detour subgraph of Q n. The main result is that for every k ≥ 2 and

Abstract. A spanning subgraph H of a graph G is a 2-detour 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, each2-detour subgraph of the n-dimensional hypercube Qn has at least c log 2 n · 2 n edges.