A unit disk graph G on a given set P of points in the plane is geometric where an edge exists between two p,q∈P if and only |pq|≤1. spanning subgraph G′ k-hop spanner for every pq∈G, there path p,q with at most k edges. We obtain following results graphs plane. Every n-vertex has 5-hop 5.5n analyze family spanners constructed by Biniaz (2020) improve upper bound number edges from 9n to 5.5n. Us...

The triangular ratio metric is studied in a domain G⊊Rn, n≥2. Several sharp bounds are proven for this metric, especially the case where unit disk of complex plane. results applied to study Hölder continuity quasiconformal mappings.

Unit disk graphs are intersection graphs of circles of unit radius in the plane. We present simple and provably good heuristics for a number of classical NP-hard optimization problems on unit disk graphs. The problems considered include maximum independent set, minimum vertex cover, minimum coloring and minimum dominating set. We also present an on-line coloring heuristic which achieves a compe...

Mobile ad hoc networks are frequently modeled by unit disk graphs. We consider several classical graph theoretic problems on unit disk graphs (Maximum Independent Set, Minimum Vertex Cover, and Minimum (Connected) Dominating Set), which are relevant to such networks. We propose two new notions for unit disk graphs: thickness and density. The thickness of a graph is the number of disk centers in...

Unit disk graphs are intersection graphs of circles of unit radius in the plane. We present simple and provably good heuristics for a number of classical NP-hard optimization problems on unit disk graphs. The problems considered include maximum independent set, minimum vertex cover, minimum coloring and minimum dominating set. We also present an on-line coloring heuristic which achieves a compe...

We provide some new sharp embeddings for p-Carleson measures and some related measures in the unit disk of the complex plane.

A disk graph is the intersection graph of disks in the plane, a unit disk graph is the intersection graph of same radius disks in the plane, and a segment graph is an intersection graph of line segments in the plane. Every disk graph can be realized by disks with centers on the integer grid and with integer radii; and similarly every unit disk graph can be realized by disks with centers on the ...

Let Φ be a strictly plurisubharmonic and radial function on the unit disk D ⊂ C and let g be the Kähler metric associated to the Kähler form ω = i 2 ∂∂̄Φ. We prove that if g is geucl-balanced of height 3 (where geucl is the standard Euclidean metric on C = R), and the function h(x) = e−Φ(z), x = |z|, extends to an entire analytic function on R, then g equals the hyperbolic metric. The proof of o...

Let n be a positive integer and λ > 0 a real number. Let Vn be a set of n points in the unit disk selected uniformly and independently at random. Define G(λ, n) to be the graph with vertex set Vn, in which two vertices are adjacent if and only if their Euclidean distance is at most λ. We call this graph a unit disk random graph. Let λ = c √ lnn/n and let X be the number of isolated points in G(...