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

Abstract.

 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 [n]={1,2,…,n} such that, for any two distinct elements x,y∈[n], there exists disjoint subsets A _x ,A _y ∈𝒮 such that x∈A _x and y∈A _y . This separating set problem can be generalized, and some bounds can be obtained from known results on families of hash functions.