We consider the connected variant of the classic mixed search game where, in
each search step, cleaned edges form a connected subgraph. We consider graph
classes with bounded connected (and monotone) mixed search number and we deal
with the question whether the obstruction set, with respect of the contraction
partial ordering, for those classes is finite. In general, there is no
guarantee that those sets are finite, as graphs are not well quasi ordered
under the contraction partial ordering relation.
In this paper we provide the obstruction set for k=2, where k is the
number of searchers we are allowed to use. This set is finite, it consists of
177 graphs and completely characterises the graphs with connected (and
monotone) mixed search number at most 2. Our proof reveals that the "sense of
direction" of an optimal search searching is important for connected search
which is in contrast to the unconnected original case. We also give a double
exponential lower bound on the size of the obstruction set for the classes
where this set is finite