We describe recent work on a variant of a distance labeling problem in graphs, called the forbidden-set labeling problem. Given a graph G = (V, E), we wish to assign labels L(x) to vertices and edges of G so that given

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We study a novel separator property called k-path separable. Roughly speaking, a k-path separable graph can be recursively separated into smaller components by sequentially removing k shortest paths. Our main result is that every minor free weighted graph is k-path separable. We then show that k-path separable graphs can be used to solve several object location problems: (1) a small-worldization with an average poly-logarithmic number of hops; (2) an (1 + ε)-approximate distance labeling scheme with O(log n) space labels; (3) a stretch-(1 + ε) compact routing scheme with tables of poly-logarithmic space; (4) an (1 + ε)-approximate distance oracle with O(n log n) space and O(log n) query time. Our results generalizes to much wider classes of weighted graphs, namely to bounded-dimension isometric sparable graphs.

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This paper studies compact routing schemes for networks with low doubling dimension. Two variants are explored, name-independent routing and labelled routing. The key results obtained for this model are the following. First, we provide the first name-independent solution. Specifically, we achieve constant stretch and polylogarithmic storage. Second, we obtain the first truly scale-free solutions, namely, the network's aspect ratio is not a factor in the stretch. Scale-free schemes are given for three problem models: name-independent routing on graphs, labelled routing on metric spaces, and labelled routing on graphs. Third, we prove a lower bound requiring linear storage for stretch < 3 schemes. This has the important ramification of separating for the first time the name-independent problem model from the labelled model, since compact stretch-1 + ε labelled schemes are known to be possible.

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Abstract. Θ k -graphs are geometric graphs that appear in the context of graph navigation. The shortest-path metric of these graphs is known to approximate the Euclidean complete graph up to a factor depending on the cone number k and the dimension of the space. TD-Delaunay graphs, a.k.a. triangular-distance Delaunay triangulations, introduced by Chew, have been shown to be plane 2-spanners of the 2D Euclidean complete graph, i.e., the distance in the TD-Delaunay graph between any two points is no more than twice the distance in the plane. Orthogonal surfaces are geometric objects defined from independent sets of points of the Euclidean space. Orthogonal surfaces are well studied in combinatorics (orders, integer programming) and in algebra. From orthogonal surfaces, geometric graphs, called geodesic embeddings can be built. In this paper, we introduce a specific subgraph of the Θ6-graph defined in the 2D Euclidean space, namely the half-Θ6-graph, composed of the evencone edges of the Θ6-graph. Our main contribution is to show that these graphs are exactly the TD-Delaunay graphs, and are strongly connected to the geodesic embeddings of orthogonal surfaces of coplanar points in the 3D Euclidean space. Using these new bridges between these three fields, we establish:-Every Θ6-graph is the union of two spanning TD-Delaunay graphs.In particular, Θ6-graphs are 2-spanners of the Euclidean graph, and the bound of 2 on the stretch factor is the best possible. It was not known that Θ6-graphs are t-spanners for some constant t, and Θ7-graphs were only known to be t-spanners for t ≈ 7.562. -Every plane triangulation is TD-Delaunay realizable, i.e., every combinatorial plane graph for which all its interior faces are triangles is the TD-Delaunay graph of some point set in the plane. Such realizability property does not hold for classical Delaunay triangulations.