Antimedian and median subsets of graphs∗

A profile on a graph G is a finite sequence of vertices of G. The remoteness of a vertex u is the sum of distances to the vertices of the profile. The set of vertices that maximize (minimize) the remoteness is the antimedian (median) set of the profile. It is proved that for every connected graph G there exists a graph H such that G is a convex subgraph of H and V (G) is the antimedian set of the profile consisting of V (H). Using linear programming it is also proved that for an arbitrary graph G and S ⊆ V (G) it can be decided in polynomial time whether S is the antimedian set of some profile. Both results are extended to median sets as well. Graphs in which every antimedian set is connected are also considered.