Let $P subset mathbb{R}^2$ be a planar $n$-point set such that each point $p in P$ has an associated radius $r_p > 0$. The transmission graph $G$ for $P$ is the directed graph with vertex set $P$ such that for any $p, q in P$, there is an edge from $p$ to $q$ if and only if $d(p, q) leq r_p$. Let $t > 1$ be a constant. A $t$-spanner for $G$ is a subgraph $H subseteq G$ with vertex set $P$ so that for any two vertices $p,q in P$, we have $d_H(p, q) leq t d_G(p, q)$, where $d_H$ and $d_G$ denote the shortest path distance in $H$ and $G$, respectively (with Euclidean edge lengths). We show how to compute a $t$-spanner for $G$ with $O(n)$ edges in $O(n (log n + log Psi))$ time, where $Psi$ is the ratio of the largest and smallest radius of a point in $P$. Using more advanced data structures, we obtain a construction that runs in $O(n log^5 n)$ time, independent of $Psi$. We give two applications for our spanners. First, we show how to use our spanner to find a BFS tree in $G$ from any given start vertex in $O(n log n)$ time (in addition to the time it takes to build the spanner). Second, we show how to use our spanner to extend a reachability oracle to answer geometric reachability queries. In a geometric reachability query we ask whether a vertex $p$ in $G$ can reach a target $q$ which is an arbitrary point in the plane (rather than restricted to be another vertex $q$ of $G$ in a standard reachability query). Our spanner allows the reachability oracle to answer geometric reachability queries with an additive overhead of $O(log nlog Psi)$ to the query time and $O(n log Psi)$ to the space.
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