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We consider the mixed search game against an agile and visible fugitive. Thisis the variant of the classic fugitive search game on graphs where searchersmay 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 alongunguarded paths. The mixed search number against an agile and visible fugitiveof a graph $G$, denoted $avms(G)$, is the minimum number of searchers requiredto capture to fugitive in this graph searching variant. Our main result is thatthis graph searching variant is monotone in the sense that the number ofsearchers required for a successful search strategy does not increase if werestrict the search strategies to those that do not permit the fugitive tovisit an already clean edge. This means that mixed search strategies against anagile and visible fugitive can be polynomially certified, and therefore thatthe 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 notionof tight bramble, that serves as an obstruction for the corresponding searchparameter. Our results imply that for a graph $G$, $avms(G)$ is equal to theCartesian tree product number of $G$ that is the minimum $k$ for which $G$ is aminor of the Cartesian product of a tree and a clique on $k$ vertices.

The mixed search game against an agile and visible fugitive is monotone