Search games with immobile hider

In this paper we discuss a type of search game with immobile hider on a graph. If the graph is weakly cyclic (i.e. no edge lies on two different cycles) we determine the value of the game and give an easy way to compute optimal strategies for the players. The value is half the length of a quasi-Euler path, i.e. a closed path of minimal length that visits all points of the network.