directed graph $G$, we let $V(G)$ and $E(G)$ denote the vertex and edge sets of $G$. We assume all edge lengths are nonnegative. For a subgraph $S \subseteq G$ (not necessarily vertex induced), let $V(S)$ denote the set of vertices of $G$, and let $E(S)$ denote the set of edges. For a subset $W \subseteq V(G)$, we define $G[W]$ to be subgraph induced by $W$. When the graph $G$ is clear from context, we let $n$ and $m$ denote $|V(G)|$ and $|E(G)|$ respectively.

For a weighted directed graph $G$ with non-negative edge lengths, we let $d_G(u, v)$ denote the (shortest path) distance from $u$ to $v$ in $G$. When the graph $G$ is clear from context, we simply denote this as $d(u, v).$ If there is no path from $u$ to $v$, we let $d(u, v) = \infty.$ When $S$ is a subgraph of $G$, we let $d_S(u, v)$ denote the (shortest path) distance from $u$ to $v$ only using the edges in $E(S).$ We denote the *roundtrip distance* between $u$ and $v$ as $d_G(u {\leftrightarrows}v) := d_G(u, v) + d_G(v, u)$ and define a roundtrip spanner.

We say that a subgraph $S \subseteq G$ is an *$\alpha$-roundtrip spanner* if 