Finding the Girth in Disk Graphs and a Directed Triangle in Transmission Graphs∗

Suppose we are given a set S ⊂ R2 of n point sites in the plane, each with an associated radius rs > 0, for s ∈ S. The disk graph D(S) for S is the undirected graph with vertex set S and an edge between s and t in S if and only if |st| ≤ rs + rt, i.e., if the disks with radius rs around s and with radius rt around t intersect. The transmission graph T (S) for S is the directed graph with vertex set S and an edge from s to t if and only if |st| ≤ rs, i.e., if the disk with radius rs around s contains the site t. We consider two problems concerning cycles in disk graphs and transmission graphs. First, we show that the weighted girth of a disk graph can be found in O(n logn) expected time, almost matching the bounds for planar graphs. Second, we present an algorithm for finding a directed triangle in a transmission graph in O(n log2 n) time. Thus, these problems are much easier for disk and transmission graphs than for general graphs.