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31
Metric graph theory and geometry: a survey
 CONTEMPORARY MATHEMATICS
"... The article surveys structural characterizations of several graph classes defined by distance properties, which have in part a general algebraic flavor and can be interpreted as subdirect decomposition. The graphs we feature in the first place are the median graphs and their various kinds of general ..."
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Cited by 44 (14 self)
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The article surveys structural characterizations of several graph classes defined by distance properties, which have in part a general algebraic flavor and can be interpreted as subdirect decomposition. The graphs we feature in the first place are the median graphs and their various kinds of generalizations, e.g., weakly modular graphs, or fibercomplemented graphs, or l1graphs. Several kinds of l1graphs admit natural geometric realizations as polyhedral complexes. Particular instances of these graphs also occur in other geometric contexts, for example, as dual polar graphs, basis graphs of (even ∆)matroids, tope graphs, lopsided sets, or plane graphs with vertex degrees and face sizes bounded from below. Several other classes of graphs, e.g., Helly graphs (as injective objects), or bridged graphs (generalizing chordal graphs), or treelike graphs such as distancehereditary graphs occur in the investigation of graphs satisfying some basic properties of the distance function, such as the Helly property for balls, or the convexity of balls or of the neighborhoods of convex sets, etc. Operators between graphs or complexes relate some of the
Collective tree spanners of graphs
 SWAT 2004
, 2004
"... In this paper we introduce a new notion of collective tree spanners. 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 d ..."
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Cited by 14 (10 self)
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In this paper we introduce a new notion of collective tree spanners. 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 any chordal graph, chordal bipartite graph or cocomparability graph admits a system of at most log 2 n collective additive tree 2–spanners and any cchordal graph admits a system of at most log 2 n collective additive tree (2⌊c/2⌋)–spanners. Towards establishing these results, we present a general property for graphs, called (α, r)– decomposition, and show that any (α, r)–decomposable graph G with n vertices admits a system of at most log 1/α n collective additive tree 2r– spanners. We discuss also an application of the collective tree spanners to the problem of designing compact and efficient routing schemes in graphs.
Tree Spanners On Chordal Graphs: Complexity And Algorithms
, 2004
"... A tree tspanner T in a graph G is a spanning tree of G such that the distance in T between every pair of vertices is at most t times their distance in G. The TREE tSPANNERprobld asks whether a graph admits a tree tspanner, given t. We ..."
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Cited by 13 (4 self)
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A tree tspanner T in a graph G is a spanning tree of G such that the distance in T between every pair of vertices is at most t times their distance in G. The TREE tSPANNERprobld asks whether a graph admits a tree tspanner, given t. We
Estimating All Pairs Shortest Paths in Restricted Graph Families: A Unified Approach (Extended Abstract)
 WG 2001, LNCS
, 2001
"... In this paper we show that a very simple and efficient approach can be used to solve the all pairs almost shortest path problem on the class of weakly chordal graphs and its different subclasses. Moreover, this approach works well also on graphs with small size of largest induced cycle and gives a u ..."
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Cited by 12 (2 self)
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In this paper we show that a very simple and efficient approach can be used to solve the all pairs almost shortest path problem on the class of weakly chordal graphs and its different subclasses. Moreover, this approach works well also on graphs with small size of largest induced cycle and gives a unified way to solve the all pairs almost shortest path and all pairs shortest path problems on different graph classes including chordal, strongly chordal, chordal bipartite, and distancehereditary graphs.
Improved Compact Routing Scheme for Chordal Graphs
 IN 16 TH INTERNATIONAL SYMPOSIUM ON DISTRIBUTED COMPUTING (DISC
, 2002
"... This paper concerns routing with succinct tables in chordal graphs. We show how to construct in polynomial time, for every nnode chordal graph, a routing scheme using routing tables and addresses of O(log³ n/ log log n) bits per node, and O(log² n/ log log n) bit not alterable headers such that ..."
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Cited by 12 (6 self)
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This paper concerns routing with succinct tables in chordal graphs. We show how to construct in polynomial time, for every nnode chordal graph, a routing scheme using routing tables and addresses of O(log³ n/ log log n) bits per node, and O(log² n/ log log n) bit not alterable headers such that the length of the route between any two nodes is at most the distance between the nodes in the graph plus two.
Additive Spanners and Distance and Routing Labeling Schemes for Hyperbolic Graphs
"... δHyperbolic metric spaces have been defined by M. Gromov in 1987 via a simple 4point condition: for any four points u, v, w, x, the two larger of the distance sums d(u, v) + d(w, x), d(u, w) + d(v, x),d(u, x) + d(v, w) differ by at most 2δ. They play an important role in geometric group theory, g ..."
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Cited by 11 (1 self)
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δHyperbolic metric spaces have been defined by M. Gromov in 1987 via a simple 4point condition: for any four points u, v, w, x, the two larger of the distance sums d(u, v) + d(w, x), d(u, w) + d(v, x),d(u, x) + d(v, w) differ by at most 2δ. They play an important role in geometric group theory, geometry of negatively curved spaces, and have recently become of interest in several domains of computer science, including algorithms and networking. For example, (a) it has been shown empirically that the internet topology embeds with better accuracy into a hyperbolic space than into an Euclidean space of comparable dimension, (b) every connected finite graph has an embedding in the hyperbolic plane so that the greedy routing based on the virtual coordinates obtained from this embedding is guaranteed to work. A connected graph G = (V, E) equipped with standard graph metric dG is δhyperbolic if the metric space (V, dG) is δhyperbolic. In this paper, using our Layering Partition technique, we provide a simpler construction of distance approximating trees of δhyperbolic graphs on n vertices with an additive error O(δ log n) and show that every nvertex δhyperbolic graph has an additive O(δ log n)spanner with at most O(δn) edges. As a consequence, we show that the family of δhyperbolic graphs with n vertices enjoys an O(δ log n)additive routing labeling scheme with O(δ log 2 n) bit labels and O(log δ) time routing protocol, and an easier constructable O(δ log n)additive distance labeling scheme with O(log 2 n) bit labels and constant time distance decoder.
Additive spanners for kchordal graphs
 In 5 th Italian Conference on Algorithms and Complexity (CIAC), volume 2653 of LNCS
, 2003
"... Abstract. In this paper we show that every chordal graph with n vertices and m edges admits an additive 4spanner with at most 2n−2 edges and an additive 3spanner with at most O(n · log n) edges. This significantly improves results of Peleg and Schäffer from [Graph Spanners, J. Graph Theory, 13(198 ..."
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Cited by 10 (3 self)
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Abstract. In this paper we show that every chordal graph with n vertices and m edges admits an additive 4spanner with at most 2n−2 edges and an additive 3spanner with at most O(n · log n) edges. This significantly improves results of Peleg and Schäffer from [Graph Spanners, J. Graph Theory, 13(1989), 99116]. Our spanners are additive and easier to construct. An additive 4spanner can be constructed in linear time while an additive 3spanner is constructable in O(m · log n) time. Furthermore, our method can be extended to graphs with largest induced cycles of length k. Any such graph admits an additive (k + 1)spanner with at most 2n − 2 edges which is constructable in O(n · k + m) time. Classification: Algorithms, Sparse Graph Spanners 1