Foxes and Hares

In the game of Foxes and Hares, we begin with \(k\) foxes, \(F_1, F_2, …, F_k\) and one hare \(H\).  Given a finite undirected graph \(G\), the foxes are first placed at \(k\) vertices of the graph, then the Hare is placed on a vertex.  The foxes and the hare alternate turns.  On the foxes’ turn, each fox may move to an adjacent vertex or remain in place, and multiple foxes may occupy the same vertex.  The hare has the same choices of moves on its turn.


The foxes win if they catch the hare, meaning there is some time at which a fox occupies the same vertex as the hare; otherwise the hare wins.  For a given graph, the minimum number of foxes required to catch the hare (regardless of the hare’s strategy) is called the graph’s fox number.

How can you find the fox number of a given graph?  What are all of the one-fox graphs?  What characteristic(s) of a graph determine(s) whether or not more than one fox is needed?  What are all of the two-fox graphs?