Let Q be a connected network with a distinguished (starting) point qo, whose arc lengths sum to one. We associate with Q a "search value" V ( Q ) representing the expected time needed for a searcher, starting at qo and moving at unit speed, to find a moving hider. We assume neither sees the other until they meet. We demonstrate that the "figure-eight" network, consisting of two equal loops joined at a central starting point, has a search value not exceeding #. This contradicts a conjecture of Gal that the search value of any network is at least 1. In the other direction, we show that V ( Q ) S 6kD for a network with k edges and diameter D.

A blind searcher and a blind hider move at below unit speed along a finite length graph Q known to both, until the first time T .when they meet. A two person zero-sum game arises if the searcher pays the hider T units. We consider circumstances under which it may be optimal for the searcher to "lie in wait" at a node of Q, hoping the hider will come to him. We also explicitly define a notion of "equilibrium in distribution" for such games, which has been implicit in the literature. We show that for the graph consisting of two nodes connected by three arcs of equal length there are optimal ambush strategies but there is no equilibrium in distribution.

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