Balanced line separators of unit disk graphs

Balanced Line Separators of Unit Disk Graphs

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...

Radial balanced metrics on the unit disk

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...

Hierarchically Specified Unit Disk Graphs

We characterize the complexity of a number of basic optimization problems for unit disk graphs speciied hierarchically as in BOW83, LW87a, Le88, LW92]. Both PSPACE-hardness results and polynomial time approximations are presented for most of the problems considered. These problems include minimum vertex coloring, maximum independent set, minimum clique cover, minimum dominating set and minimum ...

Improper colouring of (random) unit disk graphs

For any graph G, the k-improper chromatic number χ(G) is the smallest number of colours used in a colouring of G such that each colour class induces a subgraph of maximum degree k. We investigate the ratio of the kimproper chromatic number to the clique number for unit disk graphs and random unit disk graphs to extend results of McDiarmid and Reed (1999); McDiarmid (2003) (where they considered...

Improper coloring of unit disk graphs