635 research outputs found
Weighted graphs defining facets: a connection between stable set and linear ordering polytopes
A graph is alpha-critical if its stability number increases whenever an edge
is removed from its edge set. The class of alpha-critical graphs has several
nice structural properties, most of them related to their defect which is the
number of vertices minus two times the stability number. In particular, a
remarkable result of Lov\'asz (1978) is the finite basis theorem for
alpha-critical graphs of a fixed defect. The class of alpha-critical graphs is
also of interest for at least two topics of polyhedral studies. First,
Chv\'atal (1975) shows that each alpha-critical graph induces a rank inequality
which is facet-defining for its stable set polytope. Investigating a weighted
generalization, Lipt\'ak and Lov\'asz (2000, 2001) introduce critical
facet-graphs (which again produce facet-defining inequalities for their stable
set polytopes) and they establish a finite basis theorem. Second, Koppen (1995)
describes a construction that delivers from any alpha-critical graph a
facet-defining inequality for the linear ordering polytope. Doignon, Fiorini
and Joret (2006) handle the weighted case and thus define facet-defining
graphs. Here we investigate relationships between the two weighted
generalizations of alpha-critical graphs. We show that facet-defining graphs
(for the linear ordering polytope) are obtainable from 1-critical facet-graphs
(linked with stable set polytopes). We then use this connection to derive
various results on facet-defining graphs, the most prominent one being derived
from Lipt\'ak and Lov\'asz's finite basis theorem for critical facet-graphs. At
the end of the paper we offer an alternative proof of Lov\'asz's finite basis
theorem for alpha-critical graphs
Eulerian digraphs and toric Calabi-Yau varieties
We investigate the structure of a simple class of affine toric Calabi-Yau
varieties that are defined from quiver representations based on finite eulerian
directed graphs (digraphs). The vanishing first Chern class of these varieties
just follows from the characterisation of eulerian digraphs as being connected
with all vertices balanced. Some structure theory is used to show how any
eulerian digraph can be generated by iterating combinations of just a few
canonical graph-theoretic moves. We describe the effect of each of these moves
on the lattice polytopes which encode the toric Calabi-Yau varieties and
illustrate the construction in several examples. We comment on physical
applications of the construction in the context of moduli spaces for
superconformal gauged linear sigma models.Comment: 27 pages, 8 figure
Acyclic orientations with path constraints
Many well-known combinatorial optimization problems can be stated over the
set of acyclic orientations of an undirected graph. For example, acyclic
orientations with certain diameter constraints are closely related to the
optimal solutions of the vertex coloring and frequency assignment problems. In
this paper we introduce a linear programming formulation of acyclic
orientations with path constraints, and discuss its use in the solution of the
vertex coloring problem and some versions of the frequency assignment problem.
A study of the polytope associated with the formulation is presented, including
proofs of which constraints of the formulation are facet-defining and the
introduction of new classes of valid inequalities
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