Every vertex a king

A king in a tournament is a vertex which can reach every other vertex via a 1-path or 2-path. A new inductive proof is given for the existence of an n-tournament with exactly k kings for all integers n ? k ? 1 with the following exceptions: k = 2 with n arbitrary, and n = k = 4 (in which cases no such n-tournament exists). Also, given an n-tournament T, the smallest order m is determined so that there exists an m-tournament W which contains T as a subtournament and so that every vertex of W is a king. Bounds are obtained in a similar problem in which the kings of W are exactly the vertices of T.