  We consider the mixed search game against an agile and visible fugitive. This
is the variant of the classic fugitive search game on graphs where searchers
may be placed to (or removed from) the vertices or slide along edges. Moreover,
the fugitive resides on the edges of the graph and can move at any time along
unguarded paths. The mixed search number against an agile and visible fugitive
of a graph $G$, denoted $avms(G)$, is the minimum number of searchers required
to capture to fugitive in this graph searching variant. Our main result is that
this graph searching variant is monotone in the sense that the number of
searchers required for a successful search strategy does not increase if we
restrict the search strategies to those that do not permit the fugitive to
visit an already clean edge. This means that mixed search strategies against an
agile and visible fugitive can be polynomially certified, and therefore that
the problem of deciding, given a graph $G$ and an integer $k,$ whether
$avms(G)\leq k$ is in NP. Our proof is based on the introduction of the notion
of tight bramble, that serves as an obstruction for the corresponding search
parameter. Our results imply that for a graph $G$, $avms(G)$ is equal to the
Cartesian tree product number of $G$ that is the minimum $k$ for which $G$ is a
minor of the Cartesian product of a tree and a clique on $k$ vertices.
