  For vertices $u$ and $v$ of an $n$-vertex graph $G$, a $uv$-trail of $G$ is
an induced $uv$-path of $G$ that is not a shortest $uv$-path of $G$. Berger,
Seymour, and Spirkl [Discrete Mathematics 2021] gave the previously only known
polynomial-time algorithm, running in $O(n^{18})$ time, to either output a
$uv$-trail of $G$ or ensure that $G$ admits no $uv$-trail. We reduce the
complexity to the time required to perform a poly-logarithmic number of
multiplications of $n^2\times n^2$ Boolean matrices, leading to a largely
improved $O(n^{4.75})$-time algorithm.
