136 research outputs found
The diameter of type D associahedra and the non-leaving-face property
Generalized associahedra were introduced by S. Fomin and A. Zelevinsky in
connection to finite type cluster algebras. Following recent work of L. Pournin
in types and , this paper focuses on geodesic properties of generalized
associahedra. We prove that the graph diameter of the -dimensional
associahedron of type is precisely for all greater than .
Furthermore, we show that all type associahedra have the non-leaving-face
property, that is, any geodesic connecting two vertices in the graph of the
polytope stays in the minimal face containing both. This property was already
proven by D. Sleator, R. Tarjan and W. Thurston for associahedra of type .
In contrast, we present relevant examples related to the associahedron that do
not always satisfy this property.Comment: 18 pages, 14 figures. Version 3: improved presentation,
simplification of Section 4.1. Final versio
Graph properties of graph associahedra
A graph associahedron is a simple polytope whose face lattice encodes the
nested structure of the connected subgraphs of a given graph. In this paper, we
study certain graph properties of the 1-skeleta of graph associahedra, such as
their diameter and their Hamiltonicity. Our results extend known results for
the classical associahedra (path associahedra) and permutahedra (complete graph
associahedra). We also discuss partial extensions to the family of nestohedra.Comment: 26 pages, 20 figures. Version 2: final version with minor correction
Posets arising as 1-skeleta of simple polytopes, the nonrevisiting path conjecture, and poset topology
Given any polytope and any generic linear functional , one
obtains a directed graph by taking the 1-skeleton of and
orienting each edge from to for .
This paper raises the question of finding sufficient conditions on a polytope
and generic cost vector so that the graph will
not have any directed paths which revisit any face of after departing from
that face. This is in a sense equivalent to the question of finding conditions
on and under which the simplex method for linear programming
will be efficient under all choices of pivot rules. Conditions on and are given which provably yield a corollary of the desired face
nonrevisiting property and which are conjectured to give the desired property
itself. This conjecture is proven for 3-polytopes and for spindles having the
two distinguished vertices as source and sink; this shows that known
counterexamples to the Hirsch Conjecture will not provide counterexamples to
this conjecture.
A part of the proposed set of conditions is that be the
Hasse diagram of a partially ordered set, which is equivalent to requiring non
revisiting of 1-dimensional faces. This opens the door to the usage of
poset-theoretic techniques. This work also leads to a result for simple
polytopes in which is the Hasse diagram of a lattice L that the
order complex of each open interval in L is homotopy equivalent to a ball or a
sphere of some dimension. Applications are given to the weak Bruhat order, the
Tamari lattice, and more generally to the Cambrian lattices, using realizations
of the Hasse diagrams of these posets as 1-skeleta of permutahedra,
associahedra, and generalized associahedra.Comment: new results for 3-polytopes and spindles added; exposition
substantially improved throughou
Pseudograph associahedra
Given a simple graph G, the graph associahedron KG is a simple polytope whose
face poset is based on the connected subgraphs of G. This paper defines and
constructs graph associahedra in a general context, for pseudographs with loops
and multiple edges, which are also allowed to be disconnected. We then consider
deformations of pseudograph associahedra as their underlying graphs are altered
by edge contractions and edge deletions.Comment: 25 pages, 22 figure
Polytopal realizations of finite type -vector fans
This paper shows the polytopality of any finite type -vector fan,
acyclic or not. In fact, for any finite Dynkin type , we construct a
universal associahedron with the property
that any -vector fan of type is the normal fan of a
suitable projection of .Comment: 27 pages, 9 figures; Version 2: Minor changes in the introductio
Bergman Complexes, Coxeter Arrangements, and Graph Associahedra
Tropical varieties play an important role in algebraic geometry. The Bergman
complex B(M) and the positive Bergman complex B+(M) of an oriented matroid M
generalize to matroids the notions of the tropical variety and positive
tropical variety associated to a linear ideal. Our main result is that if A is
a Coxeter arrangement of type Phi with corresponding oriented matroid M_Phi,
then B+(M_Phi) is dual to the graph associahedron of type Phi, and B(M_Phi)
equals the nested set complex of A. In addition, we prove that for any
orientable matroid M, one can find |mu(M)| different reorientations of M such
that the corresponding positive Bergman complexes cover B(M), where mu(M)
denotes the Mobius function of the lattice of flats of M.Comment: 24 pages, 4 figures, new result and new proofs adde
Exchange graphs of cluster algebras have the non-leaving-face property
The claim in the title is proved
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