A connected graph is called of non-QE class if it does not admit a quadratic embedding in Euclidean space. primary contain as an isometrically embedded proper subgraph. The graphs on six vertices are completely classified into the classes QE graphs, and graphs.

A graph drawing is greedy if, for every ordered pair of vertices (x, y), there is a path from x to y such that the Euclidean distance to y decreases monotonically at every vertex of the path. Greedy drawings support a simple geometric routing scheme, in which any node that has to send a packet to a destination “greedily” forwards the packet to any neighbor that is closer to the destination than...

Let S be a set of n points and let w be a function that assigns non-negative weights to points in S. The additive weighted distance dw(p, q) between two points p, q ∈ S is defined as w(p) + d(p, q) + w(q) if p 6= q and it is zero if p = q. Here, d(p, q) denotes the (geodesic) Euclidean distance between p and q. A graph G(S,E) is called a t-spanner for the additive weighted set S of points if fo...

12:00–12:55 Registration 12:55–13:00 Opening remarks, chaired by Makoto Namiki (Toho Univ.) Yasufumi Saruwatari (Univ. of Tsukuba) Komei Fukuda (ETH Zurich) 13:00–14:20 Session 1, chaired by Yasuko Matsui (Tokai Univ.) Finding the boundary nodes of a Euclidean graph Reinhardt Euler (Univ. of Brest) Deza’s last problems Yohji Akama (Tohoku Univ.) 14:35–15:55 Session 2, chaired by Tomomi Matsui (...

A classical result of De Bruijn and Erdős shows that any given n points are either collinear or determine at least n distinct lines in the plane. Chen and Chvátal conjectured that a generalization of this result in all metric spaces, with appropriate definitions of lines, holds as well. In this note, we show that the Chen-Chvátal generalization holds for Asteroidal-Triple free graphs, where lin...

The structure of networks describing interactions between entities gives significant insights about how these systems work. Recently, an approach has been proposed to transform a graph into a collection of signals, using a multidimensional scaling technique on a distance matrix representing relations between vertices of the graph as points in a Euclidean space: coordinates are interpreted as co...

Let S be a set of n points in the plane and let G be an undirected graph with vertex set S, in which each edge (u, v) has a weight, which is equal to the Euclidean distance |uv| between the points u and v. For any two points p and q in S, we denote their shortest-path distance in G by δG(p, q). If t ≥ 1 is a real number, then we say that G is a t-spanner for S if δG(p, q) ≤ t|pq| for any two po...

We investigate a new method for partitioning a graph into two equal-sized pieces with few connecting edges. We combine ideas from two recently suggested partitioning algorithms, spectral bisection (which uses an eigenvector of a matrix associated with the graph) and geometric bisection (which applies to graphs that are meshes in Euclidean space). The new method does not require geometric coordi...