Pursuit-evasion

Pursuit-evasion (variants of which are referred to as cops and robbers and graph searching) is a family of problems in mathematics and computer science in which one group attempts to track down members of another group in an environment. Early work on problems of this type modeled the environment geometrically. In 1976, Torrence Parsons introduced a formulation whereby movement is constrained by a graph. The geometric formulation is sometimes called continuous pursuit-evasion, and the graph formulation discrete pursuit-evasion (also called graph searching). Current research is typically limited to one of these two formulations.

In the discrete formulation of the pursuit-evasion problem, the environment is modeled as a graph.

There are innumerable possible variants of pursuit-evasion, though they tend to share many elements. 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. If a pursuer occupies the same node as an evader the evader is captured and removed from the graph. The question usually posed is how many pursuers are necessary to ensure the eventual capture of all the evaders. If one pursuer suffices, the graph is called a cop-win graph. In this case, a single evader can always be captured in time linear to the number of n nodes of the graph. Capturing r evaders with k pursuers can take in the order of rn time as well, but the exact bounds for more than one pursuer are still unknown.

Often the movement rules are altered by changing the velocity of the evaders. This velocity is the maximum number of edges that an evader can move along in a single turn. In the example above, the evaders have a velocity of one. At the other extreme is the concept of infinite velocity, which allows an evader to move to any node in the graph so long as there is a path between its original and final positions that contains no nodes occupied by a pursuer. Similarly some variants arm the pursuers with "helicopters" which allow them to move to any vertex on their turn.

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