Axiomatic Characterization of the Antimedian Function on Paths and Hypercubes

An antimedian of a profile $\\pi = (x_1, x_2, \\ldots , x_k)$ of vertices of a graph $G$ is a vertex maximizing the sum of the distances to the elements of the profile. The antimedian function is defined on the set of all profiles on $G$ and has as output the set of antimedians of a profile. It is a typical location function for finding a location for an obnoxious facility. The `converse' of the antimedian function is the median function, where the distance sum is minimized. The median function is well studied. For instance it has been characterized axiomatically by three simple axioms on median graphs. The median function behaves nicely on many classes of graphs. In contrast the antimedian function does not have a nice behavior on most classes. So a nice axiomatic characterization may not be expected. In this paper such a characterization is obtained for the two classes of graphs on which the antimedian is well-behaved: paths and hypercubes

A simple axiomatization of the median procedure on median graphs

A profile = (x1, ..., xk), of length k, in a finite connected graph G is a sequence of vertices of G, with repetitions allowed. A median x of is a vertex for which the sum of the distances from x to the vertices in the profile is minimum. The median function finds the set of all medians of a profile. Medians are important in location theory and consensus theory. A median graph is a graph for which every profile of length 3 has a unique median. Median graphs are well studied. They arise in many arenas, and have many applications. We establish a succinct axiomatic characterization of the median procedure on median graphs. This is a simplification of the characterization given by McMorris, Mulder and Roberts [17] in 1998. We show that the median procedure can be characterized on the class of all median graphs with only three simple and intuitively appealing axioms: anonymity, betweenness and consistency. We also extend a key result of the same paper, characterizing the median function for profiles of even length on median graphs

Axiomatic Characterization of the Median and Antimedian Function on a Complete Graph minus a Matching

__Abstract__ A median (antimedian) of a profile of vertices on a graph G is a vertex that minimizes (maximizes) the sum of the distances to the elements in the profile. The median (antimedian) function has as output the set of medians (antimedians) of a profile. It is one of the basic models for the location of a desirable (obnoxious) facility in a network. The median function is well studied. For instance it has been characterized axiomatically by three simple axioms on median graphs. The median function behaves nicely on many classes of graphs. In contrast the antimedian function does not have a nice behavior on most classes. So a nice axiomatic characterization may not be expected. In this paper an axiomatic characterization is obtained for the median and antimedian function on complete graphs minus a matching

Five axioms for location functions on median graphs

__Abstract__ In previous work, two axiomatic characterizations were given for the median function on median graphs: one involving the three simple and natural axioms anonymity, betweenness and consistency; the other involving faithfulness, consistency and ½-Condorcet. To date, the independence of these axioms has not been a serious point of study. The aim of this paper is to provide the missing answers. The independent subsets of these five axioms are determined precisely and examples provided in each case on arbitrary median graphs. There are three cases that stand out. Here non-trivial examples and proofs are needed to give a full answer. Extensive use of the structure of median graphs is used throughout

Axioms for consensus functions on the n-cube

An elementary general result is proved that allows for simple characterizations of well-known location/consensus functions (median, mean and center) on the n-cube. In addition, alternate new characterizations are given for the median and anti-median functions on the n-cube.Comment: 12 page