Axiomatic characterization of the interval function of a graph

A fundamental notion in metric graph theory is that of the interval function I : V × V → 2V – {∅} of a (finite) connected graph G = (V,E), where I(u,v) = { w | d(u,w) + d(w,v) = d(u,v) } is the interval between u and v. An obvious question is whether I can be characterized in a nice way amongst all functions F : V × V -> 2V – {∅}. This was done in [13, 14, 16] by axioms in terms of properties of the functions F. The authors of the present paper, in the conviction that characterizing the interval function belongs to the central questions of metric graph theory, return here to this result again. In this characterization the set of axioms consists of five simple, and obviously necessary, axioms, already presented in [9], plus two more complicated axioms. The question arises whether the last two axioms are really necessary in the form given or whether simpler axioms would do the trick. This question turns out to be non-trivial. The aim of this paper is to show that these two supplementary axioms are optimal in the following sense. The functions satisfying only the five simple axioms are studied extensively. Then the obstructions are pinpointed why such functions may not be the interval function of some connected graph. It turns out that these obstructions occur precisely when either one of the supplementary axioms is not satisfied. It is also shown that each of these supplementary axioms is independent of the other six axioms. The presented way of proving the characterizing theorem (Theorem 3 here) allows us to find two new separate ``intermediate'' results (Theorems 1 and 2). In addition some new characterizations of modular and median graphs are presented. As shown in the last section the results of this paper could provide a new perspective on finite connected graphs

The interval function of a connected graph and a characterization of geodetic graphs

summary:The interval function (in the sense of H. M. Mulder) is an important tool for studying those properties of a connected graph that depend on the distance between vertices. An axiomatic characterization of the interval function of a connected graph was published by Nebeský in 1994. In Section 2 of the present paper, a simpler and shorter proof of that characterization will be given. In Section 3, a characterization of geodetic graphs will be established; this characterization will utilize properties of the interval function

Means and covariance functions for geostatistical compositional data: an axiomatic approach

This work focuses on the characterization of the central tendency of a sample of compositional data. It provides new results about theoretical properties of means and covariance functions for compositional data, with an axiomatic perspective. Original results that shed new light on the geostatistical modeling of compositional data are presented. As a first result, it is shown that the weighted arithmetic mean is the only central tendency characteristic satisfying a small set of axioms, namely continuity, reflexivity and marginal stability. Moreover, this set of axioms also implies that the weights must be identical for all parts of the composition. This result has deep consequences on the spatial multivariate covariance modeling of compositional data. In a geostatistical setting, it is shown as a second result that the proportional model of covariance functions (i.e., the product of a covariance matrix and a single correlation function) is the only model that provides identical kriging weights for all components of the compositional data. As a consequence of these two results, the proportional model of covariance function is the only covariance model compatible with reflexivity and marginal stability

Axiomatic characterization of the absolute median on cube-free median networks

In Vohra, European J. Operational Research 90 (1996) 78 – 84, a characterization of the absolute median of a tree network using three simple axioms is presented. This note extends that result from tree networks to cube-free median networks. A special case of such networks is the grid structure of roads found in cities equipped with the Manhattan metric.