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 $uv\in E$. It is known that every $n$-vertex graph admits a $(2k-1)$-spanner with $O(n^{1+1/k})$ edges for $k\geq 1$. This bound is the best possible for $1\leq k\leq 9$ and is conjectured to be optimal due to Erd\H{o}s' girth conjecture. We study $t$-spanners for $t\in {2,3}$ for geometric intersection graphs in the plane. These spanners are also known as \emph{$t$-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.