HOP-SPANNERS FOR GEOMETRIC INTERSECTION GRAPHS

A t-spanner of a graph G = (V, E) is a subgraph H = (V, E') that contains a uv-path of length at most t for every uv is an element of E. It is known that every n-vertex graph admits a (2k - 1)-spanner with O(n(1+1/k)) edges for k >= 1. This bound is the best possible for 1 < k < 9 and is conjectured to be optimal due to Erdos' girth conjecture.We study t-spanners for t E {2, 3} for geometric intersection graphs in the plane. These spanners are also known as t-hop spanners to emphasize the use of graph-theoretic distances (as opposed to Euclidean distances between the geometric objects or their centers). We obtain the following results: (1) Every n-vertex unit disk graph (UDG) admits a 2 -hop spanner with O(n) edges; improving upon the previous bound of O(n log n). (2) The intersection graph of n axis-aligned fat rectangles admits a 2-hop spanner with O(n log n) edges, and this bound is tight up to a factor of log log n. (3) The intersection graph of n fat convex bodies in the plane admits a 3-hop spanner with O(n log n) edges. (4) The intersection graph of n axis-aligned rectangles admits a 3-hop spanner with O(n log2 n) edges.