Geodesics in Graphs, an Extremal Set Problem, and Perfect Hash Families

A set U of vertices of a graph G is called a geodetic set if the union of all the geodesics joining pairs of points of U is the whole graph G. One result in this paper is a tight lower bound on the minimum number of vertices in a geodetic set. In order to obtain that result, the following extremal set problem is solved. Find the minimum cardinality of a collection S of subsets of 1⁄2n 1⁄4 f1; 2; . . . ; ng such that, for any two distinct elements x; y 2 1⁄2n , there exists disjoint subsets Ax;Ay 2 S such that x 2 Ax and y 2 Ay . This separating set problem can be generalized, and some bounds can be obtained from known results on families of hash functions.