Here we are given a finite connected undirected graph (V,E) (so V is the set of vertices and E the set of edges) and weights on the edges, i.e., c(e) ∈ R for all e ∈ E. The task is to find a set T ⊆ E such that (V, T ) is a (spanning) tree and ∑ e∈T c(e) is minimum. (Recall that a tree is a connected graph without cycles.) The figure below shows on the left a set V of eight points in the Euclid...

In this note, we characterize the graphs (1-skeletons) of some piecewise Euclidean simplicial and cubical complexes having nonpositive curvature in the sense of Gromov’s CAT(0) inequality. Each such cell complex K is simply connected and obeys a certain flag condition. It turns out that if, in addition, all maximal cells are either regular Euclidean cubes or right Euclidean triangles glued in a...

Let V be a set of distinct points in the Euclidean plane. For each point x 2 V , let sx be the ball centered at x with radius equal to the distance from x to its nearest neighbour. We refer to these balls as the spheres of in uence of the set V . The sphere of in uence graph on V is de ned as the graph where (x; y) is an edge if and only if sx and sy intersect. In this extended abstract, we dem...

An Euclidean greedy embedding of a graph is a straight-line embedding in the plane, such that for every pair of vertices s and t, the vertex s has a neighbor v with smaller distance to t than s. This drawing style is motivated by greedy geometric routing in wireless sensor networks. A Christmas cactus is a connected graph in which every two simple cycles have at most one vertex in common and in...

We consider the problem of locally constructing a spanning subgraph that approximates the Euclidean minimum spanning tree of a unit disk graph. We show that for any k ≥ 2 there exists a klocalized distributed algorithm that, given a unit disk graph U in the plane, constructs a planar subgraph of U containing a Euclidean MST on V (U), whose degree is at most 6, and whose total weight is at most ...

The point placement problem is to determine the positions of a linear set of points, P = {p1, p2, p3, . . . , pn}, uniquely, up to translation and reflection, from the fewest possible distance queries between pairs of points. Each distance query corresponds to an edge in a graph, called point placement graph (ppg), whose vertex set is P . The uniqueness requirement of the placement translates t...

We address the chromatic number of a class of Euclidean distance graphs having vertex set Q, the 3-dimensional rational space. It is shown that if s is any positive integer with no prime factors congruent to 2 (mod 3), and G is the graph with vertex set Q where any two vertices are adjacent if and only if they are distance √ 2s apart, then G has chromatic number 4. Along the way, we obtain a fe...

The unit distance graph R is the graph with vertex set R2 in which two vertices (points in the plane) are adjacent if and only if they are at Euclidean distance 1. We prove that the circular chromatic number of R is at least 4, thus “tightening” the known lower bound on the chromatic number of R.

The vertices of the odd-distance graph are the points of the plane R. Two points are connected by an edge if their Euclidean distance is an odd integer. We prove that the chromatic number of this graph is at least five. We also prove that the odd-distance graph in R is countably choosable, while such a graph in R is not.

Many tractable algorithms for solving the Constraint Satisfaction Problem (Csp) have been developed using the notion of the treewidth of some graph derived from the input Csp instance. In particular, the incidence graph of the Csp instance is one such graph. We introduce the notion of an incidence graph for modal logic formulae in a certain normal form. We investigate the parameterized complexi...