Axiomatic characterization of the center function. The case of non-universal axioms

The center function is defined on a connected graph G, where the input is any finite sequence of vertices of G and the output is the set of all vertices that minimize the maximum distance to the entries of the input. If the input is a sequence containing each vertex of G once, then the output is just the classical center of G. In the axiomatic approach, one wants to establish a set of properties or consensus axioms that characterize the function. We refer to an axiom as a universal axiom if the center function satisfies this axiom on any connected graph. In a previous paper, our focus was on classes of graphs on which we were able to characterize the center function in terms of universal axioms. In this paper, the focus is on classes of graphs on which these universal axioms do not characterize the center function. We introduce non-universal axioms that, together with some universal axioms, provide new characterizations of the center function: on cocktail party graphs, on complete bipartite graphs, and on block graphs.