The simplest model of a wireless network graph is the Unit Disk Graph (UDG): an edge exists in UDG if the Euclidean distance between its endpoints is ≤ 1. The problem of constructing planar spanners of Unit Disk Graphs with respect to the Euclidean distance has received considerable attention from researchers in computational geometry and ad-hoc wireless networks. In this paper, we present an a...

We investigate the following problem proposed by Alcatel. A satellite sends information to receivers on earth, each of which is listening on a chosen frequency. Technically, it is impossible for the satellite to precisely focus its signal onto a receiver. Part of the signal will be spread in an area around its destination and this creates noise for nearby receivers on the same frequency. Howeve...

The contact graph of an arbitrary finite packing of unit balls in Euclidean 3-space is the (simple) graph whose vertices correspond to the packing elements and whose two vertices are connected by an edge if the corresponding two packing elements touch each other. One of the most basic questions on contact graphs is to find the maximum number of edges that a contact graph of a packing of n unit ...

The quadrance between two points A1 = (x1, y1) and A2 = (x2, y2) is the number Q(A1, A2) = (x1 − x2) + (y1 − y2). Let q be an odd prime power and Fq be the finite field with q elements. The unit-quadrance graph Dq has the vertex set F 2 q , and X,Y ∈ F 2 q are adjacent if and only if Q(A1, A2) = 1. In this paper, we study some colouring problems for the unit-quadrance graph Dq and discuss some ...

We prove a geometric version of the graph separator theorem for the unit disk intersection graph: for any set of n unit disks in the plane there exists a line ` such that ` intersects at most O( √ (m+ n) logn) disks and each of the halfplanes determined by ` contains at most 2n/3 unit disks from the set, where m is the number of intersecting pairs of disks. We also show that an axis-parallel li...

We say any order ~ is a tolerance order on a set of vertices if we may assign to each vertex x an interval Ix of real numbers and a real number tx called a tolerance in such a way that x~,y if and only if the overlap of Ix and ly is less than the minimum of t~ and ty and the center of I~ is less than the center of Iy. An order is a bitolerance order if and only if there are intervals Ix and rea...

We present one more proof of the fact that the class of proper interval graphs is precisely the class of unit interval graphs. The proof leads to a new and efficient O(n) time and space algorithm that transforms a proper interval model of the graph into a unit model, where all the extremes are integers in the range 0 to O(n2), solving a problem posed by Gardi (Discrete Math., 307 (22), 2906–290...

Let S ⊂ R be a set of point sites. The unit disk graph UD(S) of S has vertex set S and an edge between two sites s, t if and only if |st| ≤ 1. We present a data structure that maintains the connected components of UD(S) when S changes dynamically. It takes O(log n) time to insert or delete a site in S and O(log n/ log log n) time to determine if two sites are in the same connected component. He...

The minimum clique partition (MCP) problem is that of partitioning the vertex set of a given graph into a minimum number of cliques. Given n points in the plane, the corresponding unit disk graph (UDG) has these points as vertices, and edges connecting points at distance at most 1. MCP in unit disk graphs is known to be NP-hard and several constant factor approximations are known, including a r...