Finding Triangles and Computing the Girth in Disk Graphs∗

Let S ⊂ R be a set of n point sites, where each s ∈ S has an associated radius rs > 0. The disk graph D(S) of S is the graph with vertex set S and an edge between two sites s and t if and only if |st| ≤ rs + rt, i.e., if the disks with centers s and t and radii rs and rt, respectively, intersect. Disk graphs are useful to model sensor networks. We study the problems of finding triangles and of computing the girth in disk graphs. These problems are notoriously hard for general graphs, but better solutions exist for special graph graph classes, such as planar graphs. We obtain similar results for disk graphs. In particular, we observe that the unweighted girth of a disk graph can be computed in O(n log n) worst-case time and that a shortest (Euclidean) triangle in a disk graph can be found in O(n log n) expected time.