  In this paper we analyze a variant of the pursuit-evasion game on a graph $G$
where the intruder occupies a vertex, is allowed to move to adjacent vertices
or remain in place, and is 'invisible' to the searcher, meaning that the
searcher operates with no knowledge of the position of the intruder. On each
stage, the searcher is allowed to inspect an arbitrary set of $k$ vertices. The
minimum $k$ for which the searcher can guarantee the capture of the intruder is
called the inspection number of $G$. We also introduce and study the
topological inspection number, a quantity that captures the limiting behavior
of the inspection number under subdivisions of $G$. Our central theorem
provides a full classification of graphs with topological inspection number up
to $3$.
