6 research outputs found
Obstructions to shellability, partitionability, and sequential Cohen-Macaulayness
For a property of simplicial complexes, a simplicial complex
is an obstruction to if itself does not satisfy
but all of its proper restrictions satisfy . In this paper, we
determine all obstructions to shellability of dimension , refining the
previous work by Wachs. As a consequence we obtain that the set of obstructions
to shellability, that to partitionability and that to sequential
Cohen-Macaulayness all coincide for dimensions . We also show that these
three sets of obstructions coincide in the class of flag complexes. These
results show that the three properties, hereditary-shellability,
hereditary-partitionability, and hereditary-sequential Cohen-Macaulayness are
equivalent for these classes
Unique Sink Orientations of Grids
We introduce unique sink orientations of grids as digraph models for many well-studied problems, including linear programming over products of simplices, generalized linear complementarity problems over P-matrices (PGLCP), and simple stochastic games. We investigate the combinatorial structure of such orientations and develop randomized algorithms for finding the sink. We show that the orientations arising from PGLCP satisfy the Holt-Klee condition known to hold for polytope digraphs, and we give the first expected linear-time algorithms for solving PGLCP with a fixed number of block
Strongly Signable and Partitionable Posets
The class of Strongly Signable partially ordered sets is introduced and studied. It is show that strong signability, reminiscent of Björner–Wachs ’ recursive coatom orderability, provides a useful and broad sufficient condition for a poset to be dual CR and hence partitionable. The flag h-vectors of strongly signable posets are therefore non-negative. It is proved that recursively shellable posets, polyhedral fans, and face lattices of partitionable simplicial complexes are all strongly signable, and it is conjectured that all spherical posets are. It is concluded that the barycentric subdivision of a partitionable complex is again partitionable, and an algorithm for producing a partitioning of the subdivision from a partitioning of the complex is described. An expression for the flag h-polynomial of a simplicial complex in terms of its h-vector is given, and is used to demonstrate that the flag h-vector is symmetric or non-negative whenever the h-vector is