A typical, basic example is as follows (cops and robber games): Pursuers and evaders occupy nodes of a graph.
The two sides take alternate turns, which consist of each member either staying put or moving along an edge to an adjacent node.
Other variants ignore the restriction that pursuers and evaders must always occupy a node and allow for the possibility that they are positioned somewhere along an edge.
The complexity of several pursuit–evasion variants, namely how many pursuers are needed to clear a given graph and how a given number of pursuers should move on the graph to clear it with either a minimum sum of their travel distances or minimum task-completion time, has been studied by Nimrod Megiddo, S. L. Hakimi, Michael R. Garey, David S. Johnson, and Christos H. Papadimitriou (J. ACM 1988), and R. Borie, C. Tovey and S.
[4] Solving multi-player pursuit–evasion games has also received increased attention; see R Vidal et al., Chung and Furukawa [1], Hespanha et al. and the references therein.
In the continuous formulation of pursuit–evasion games, the environment is modeled geometrically, typically taking the form of the Euclidean plane or another manifold.
Variants of the game may impose maneuverability constraints on the players, such as a limited range of speed or acceleration.
[5] One of the initial applications of the pursuit–evasion problem was missile guidance systems formulated by Rufus Isaacs at the RAND Corporation.