For a median graph G and a vertex v of G that is not a cut-vertex we show that G − v is a median graph precisely when v is not the center of a bipartite wheel, which is in turn equivalent with the existence of a certain edge elimination scheme for edges incident with v. This implies a characterization of vertex-critical (respectively, vertex-complete) median graphs, which are median graphs whos...

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 12 -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. T...

In this paper, we introduce a new mesh filtering method: a mesh median filter. This is an application of the median filter to smoothen 3-D noisy shapes given by triangle meshes. An algorithm of the mesh median filter is realized by applying the median filter to face normals on triangle meshes and updating mesh vertex positions to make them fit to the filtered normals. As an advanced modificatio...

In a tree one can find the median set of a profile simply by starting at an arbitrary vertex and then moving to the majority of the profile. This strategy is formulated for arbitrary graphs. The graphs for which this strategy produces always the median set M(E), for each profile 7t, are precisely the median graphs. AMS C/ass$cication: Primary: 05C12,OSC75,05C99; secondary: 90B80

Ensemble clustering is a recently evolving research direction in cluster analysis and has found several different application domains. In this work the complex ensemble clustering problem is reduced to the well-known Euclidean median problem by clustering embedding in vector spaces. The Euclidean median problem is solved by the Weiszfeld algorithm and an inverse transformation maps the Euclidea...

In this paper we consider two conjectures of H.M. Mulder [M] and give partial solutions for them. First we prove that a triangle free superspherical graph is interval regular provided it satisfies an additional (not too strong) condition. Furthermore, we show that a spherical interval regular graph is interval monotone.

For a multi-set E of splits {bipartitions) of a finite set X, we introduce the multi-split graph G(E). This graph Is a natural extension of the Buneman graph. Indeed, it is shown that several results pertaining to the Buneman graph extend to the multi-split graph. In addition, in case E is derived from a set R of partitions of X by taking parts together with their complements, we show that the ...

An antimedian of a profile π = (x1, x2, . . . , xk) 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 f...

A median (antimedian) of a profile of vertices on a graph G is a vertex that minimizes (maximizes) the remoteness value, that is, the sum of the distances to the elements in the profile. The median (or 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 (or obnoxious) facility in a network. The median fu...

Introduction. The geometric median is a classic robust estimator of centrality for data in Euclidean spaces, and it has been generalized in analytical manifold in [1]. Recently, an intrinsic Riemannian framework for Orientation Distribution Function (ODF) was proposed for the calculation in ODF field [2]. In this work, we prove the unique existence of the Riemannian median in ODF space. Then we...