  We study the classical problem introduced by R. Isaacs and S. Gal of
minimizing the time to find a hidden point $H$ on a network $Q$ moving from a
known starting point. Rather than adopting the traditional continuous unit
speed path paradigm, we use the ``expanding search'' paradigm recently
introduced by the authors. Here the regions $S\left( t\right) $ that have been
searched by time $t$ are increasing from the starting point and have total
length $t$. Roughly speaking the search follows a sequence of arcs $a_{i}$ such
that each one starts at some point of an earlier one. This type of search is
often carried out by real life search teams in the hunt for missing persons,
escaped convicts, terrorists or lost airplanes. The paper which introduced this
type of search solved the adversarial problem (where $H$ is hidden to take a
long time to find) for the cases where $Q$ is a tree or is 2-arc-connected.
This paper solves the game on some additional families of networks. However the
main contribution is to give strategy classes which can be used on any network
and have expected search times which are within a factor close to 1 of the
value of the game (minimax search time). We identify cases where our strategies
are in fact optimal.
