  We consider a search and rescue game introduced recently by the first author.
An immobile target or targets (for example, injured hikers) are hidden on a
graph. The terrain is assumed to dangerous, so that when any given vertex of
the graph is searched, there is a certain probability that the search will come
to an end, otherwise with the complementary {\em success probability} the
search can continue. A Searcher searches the graph with the aim of finding all
the targets with maximum probability. Here, we focus on the game in the case
that the graph is a cycle. In the case that there is only one target, we solve
the game for equal success probabilities, and for a class of games with unequal
success probabilities. For multiple targets and equal success probabilities, we
give a solution for an adaptive Searcher and a solution in a special case for a
non-adaptive Searcher. We also consider a continuous version of the model,
giving a full solution for an adaptive Searcher and approximately optimal
solutions in the non-adaptive case.
