In this paper we show that the θ-graph with 4 cones has constant stretch factor, i.e., there is a path between any pair of vertices in this graph whose length is at most a constant times the Euclidean distance between that pair of vertices. This is the last θ-graph for which it was not known whether its stretch factor was bounded.

A non-complete geometric distance-regular graph is the point graph of a partial geometry in which the set of lines is a set of Delsarte cliques. In this paper, we prove that for fixed integer m ≥ 2, there are only finitely many non-geometric distance-regular graphs with smallest eigenvalue at least −m, diameter at least three and intersection number c2 ≥ 2.

For a set S of n lines labeled from 1 to n, we say that S supports an n-vertex planar graph G if for every labeling from 1 to n of its vertices, G has a straight-line crossing-free drawing with each vertex drawn as a point on its associated line. It is known from previous work [4] that no set of n parallel lines supports all n-vertex planar graphs. We show that intersecting lines, even if they ...

Greedy embedding (or drawing) is a simple and efficient strategy to route messages in wireless sensor networks. For each sourcedestination pair of nodes s, t in a greedy embedding there is always a neighbor u of s that is closer to t according to some distance metric. The existence of Euclidean greedy embeddings in R is known for certain graph classes such as 3-connected planar graphs. We compl...

An important application of distance geometry to biochemistry studies the embeddings of the vertices of a weighted graph in the three-dimensional Euclidean space such that the edge weights are equal to the Euclidean distances between corresponding point pairs. When the graph represents the backbone of a protein, one can exploit the natural vertex order to show that the search space for feasible...

A b s t r a c t. Chew and Dobkin et. al. have shown that the Delaunay triangulation and its variants are sparse approximations of the complete graph, in that the shortest distance between two sites within the triangulation is bounded by a constant multiple of their Euclidean separation. In this paper, we show that other classical triangulation algorithms, such as the greedy triangulation, and m...

For a graph whose vertex set is finite of points in the Euclidean $$d$$ -space consider closed (open) balls with diameters induced by its edges. The called (an open) Tverberg if these intersect. Using idea halving lines, we show that (i) for any plane, there exists Hamiltonian cycle graph; (ii) $$ n red and blue perfect red-blue matching graph. Also, prove (iii) even -space, an open (iv)

Graph Isomorphism is a widely studied problem due to its practical applications in various fields of networks, chemistry and finger print detection, recent problems in biology such as diabetes detection, protein structure and information retrieval. An approach to the graph isomorphism detection is based on vertex invariant. In the existing approach vertex invariants is used to partition the mat...

An important application of distance geometry to biochemistry studies the embeddings of the vertices of a weighted graph in the three-dimensional Euclidean space such that the edge weights are equal to the Euclidean distances between corresponding point pairs. When the graph represents the backbone of a protein, one can exploit the natural vertex order to show that the search space for feasible...