671 research outputs found
On a generalization of median graphs: -median graphs
Median graphs are connected graphs in which for all three vertices there is a
unique vertex that belongs to shortest paths between each pair of these three
vertices. To be more formal, a graph is a median graph if, for all , it holds that where
denotes the set of all vertices that lie on shortest paths connecting
and . In this paper we are interested in a natural generalization of
median graphs, called -median graphs. A graph is a -median graph, if
there are vertices such that, for all , it holds that , . By definition, every median graph with vertices is an -median graph.
We provide several characterizations of -median graphs that, in turn, are
used to provide many novel characterizations of median graphs
Groups acting on quasi-median graphs. An introduction
Quasi-median graphs have been introduced by Mulder in 1980 as a
generalisation of median graphs, known in geometric group theory to naturally
coincide with the class of CAT(0) cube complexes. In his PhD thesis, the author
showed that quasi-median graphs may be useful to study groups as well. In the
present paper, we propose a gentle introduction to the theory of groups acting
on quasi-median graphs.Comment: 16 pages. Comments are welcom
Folding median graphs
Extending Stallings' foldings of trees, we show in this article that every
parallel-preserving map between median graphs factors as an isometric embedding
through a sequence of elementary transformations which we call foldings and
swellings. This new construction proposes a unified point of view on Beeker and
Lazarovich's work on folding pocsets and on Ben-Zvi, Kropholler, and Lyman's
work on folding nonpositively curved cube complexes.Comment: 45 pages, 5 figures. Comments are welcome
Condorcet Domains, Median Graphs and the Single Crossing Property
Condorcet domains are sets of linear orders with the property that, whenever
the preferences of all voters belong to this set, the majority relation has no
cycles. We observe that, without loss of generality, such domain can be assumed
to be closed in the sense that it contains the majority relation of every
profile with an odd number of individuals whose preferences belong to this
domain.
We show that every closed Condorcet domain is naturally endowed with the
structure of a median graph and that, conversely, every median graph is
associated with a closed Condorcet domain (which may not be a unique one). The
subclass of those Condorcet domains that correspond to linear graphs (chains)
are exactly the preference domains with the classical single crossing property.
As a corollary, we obtain that the domains with the so-called `representative
voter property' (with the exception of a 4-cycle) are the single crossing
domains.
Maximality of a Condorcet domain imposes additional restrictions on the
underlying median graph. We prove that among all trees only the chains can
induce maximal Condorcet domains, and we characterize the single crossing
domains that in fact do correspond to maximal Condorcet domains.
Finally, using Nehring's and Puppe's (2007) characterization of monotone
Arrowian aggregation, our analysis yields a rich class of strategy-proof social
choice functions on any closed Condorcet domain
Sparse Median Graphs Estimation in a High Dimensional Semiparametric Model
In this manuscript a unified framework for conducting inference on complex
aggregated data in high dimensional settings is proposed. The data are assumed
to be a collection of multiple non-Gaussian realizations with underlying
undirected graphical structures. Utilizing the concept of median graphs in
summarizing the commonality across these graphical structures, a novel
semiparametric approach to modeling such complex aggregated data is provided
along with robust estimation of the median graph, which is assumed to be
sparse. The estimator is proved to be consistent in graph recovery and an upper
bound on the rate of convergence is given. Experiments on both synthetic and
real datasets are conducted to illustrate the empirical usefulness of the
proposed models and methods
Two relations for median graphs
AbstractWe generalize the well-known relation for trees n−m=1 to the class of median graphs in the following way. Denote by qi the number of subgraphs isomorphic to the hypercube Qi in a median graph. Then, ∑i⩾0(−1)iqi=1. We also give an explicit formula for the number of Θ-classes in a median graph as k=−∑i⩾0(−1)iiqi
The structure of median graphs
AbstractA median graph is a connected graph, such that for any three vertices u,ν and w there is exactly one vertex x that lies simultaneously on a shortest (u, ν)-path a shortest (ν, w)-path and a shortest (w, u)-path. It is proved that a median graph can be obtained from a one-vertex graph by an expansion procedure. From this characterization some nice properties are derived
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