The median function on graphs with bounded profiles

The median of a profile @p=(u"1,...,u"k) of vertices of a graph G is the set of vertices x that minimize the sum of distances from x to the vertices of @p. It is shown that for profiles @p with diameter @q the median set can be computed within an isometric subgraph of G that contains a vertex x of @p and the r-ball around x, where r>[email protected]@q/|@p|. The median index of a graph and r-joins of graphs are introduced and it is shown that r-joins preserve the property of having a large median index. Consensus strategies are also briefly discussed on a graph with bounded profiles.