57 research outputs found
Tree-width and dimension
Over the last 30 years, researchers have investigated connections between
dimension for posets and planarity for graphs. Here we extend this line of
research to the structural graph theory parameter tree-width by proving that
the dimension of a finite poset is bounded in terms of its height and the
tree-width of its cover graph.Comment: Updates on solutions of problems and on bibliograph
A study of discrepancy results in partially ordered sets
In 2001, Fishburn, Tanenbaum, and Trenk published a pair of papers that introduced the notions of linear and weak discrepancy of a partially ordered set or poset. Linear discrepancy for a poset is the least k such that for any ordering of the points in the poset there is a pair of incomparable points at least distance k away in the ordering. Weak discrepancy is similar to linear discrepancy except that the distance is observed over weak labelings (i.e. two points can have the same label if they are incomparable, but order is still preserved). My thesis gives a variety of results pertaining to these properties and other forms of discrepancy in posets. The first chapter of my thesis partially answers a question of Fishburn, Tanenbaum, and Trenk that was to characterize those posets with linear discrepancy two. It makes the characterization for those posets with width two and references the paper where the full characterization is given. The second chapter introduces the notion of t-discrepancy which is similar to weak discrepancy except only the weak labelings with
at most t copies of any label are considered. This chapter shows that determining a poset's t-discrepancy is NP-Complete. It also gives the t-discrepancy for the disjoint sum of chains and provides a polynomial time algorithm for determining t-discrepancy of semiorders. The third chapter presents another notion of discrepancy namely total discrepancy which minimizes the average distance between incomparable elements. This chapter proves that finding this value can be done in polynomial time unlike linear discrepancy and t-discrepancy. The final chapter answers another question of Fishburn, Tanenbaum, and Trenk that asked to characterize those posets that have equal linear and weak discrepancies. Though determining the answer of whether the weak discrepancy and linear discrepancy of a poset are equal is an NP-Complete problem, the set of minimal posets that have this property are given. At the end of the thesis I discuss two other open problems not mentioned in the previous chapters that relate to linear discrepancy. The first asks if there is a link between a poset's dimension and its linear discrepancy. The second refers to approximating linear discrepancy and possible ways to do it.Ph.D.Committee Chair: Trotter, William T.; Committee Member: Dieci, Luca; Committee Member: Duke, Richard; Committee Member: Randall, Dana; Committee Member: Tetali, Prasa
Fiber polytopes for the projections between cyclic polytopes
The cyclic polytope is the convex hull of any points on the
moment curve in . For , we
consider the fiber polytope (in the sense of Billera and Sturmfels) associated
to the natural projection of cyclic polytopes which
"forgets" the last coordinates. It is known that this fiber polytope has
face lattice indexed by the coherent polytopal subdivisions of which
are induced by the map . Our main result characterizes the triples
for which the fiber polytope is canonical in either of the following
two senses:
- all polytopal subdivisions induced by are coherent,
- the structure of the fiber polytope does not depend upon the choice of
points on the moment curve.
We also discuss a new instance with a positive answer to the Generalized
Baues Problem, namely that of a projection where has only
regular subdivisions and has two more vertices than its dimension.Comment: 28 pages with 1 postscript figur
Forbidden subposet problems in the grid
For posets and , extremal and saturation problems about weak and
strong -free subposets of have been studied mostly in the case is
the Boolean poset , the poset of all subsets of an -element set ordered
by inclusion. In this paper, we study some instances of the problem with
being the grid, and its connections to the Boolean case and to the forbidden
submatrix problem
The leafage of a chordal graph
The leafage l(G) of a chordal graph G is the minimum number of leaves of a
tree in which G has an intersection representation by subtrees. We obtain upper
and lower bounds on l(G) and compute it on special classes. The maximum of l(G)
on n-vertex graphs is n - lg n - (1/2) lg lg n + O(1). The proper leafage l*(G)
is the minimum number of leaves when no subtree may contain another; we obtain
upper and lower bounds on l*(G). Leafage equals proper leafage on claw-free
chordal graphs. We use asteroidal sets and structural properties of chordal
graphs.Comment: 19 pages, 3 figure
The polytope of non-crossing graphs on a planar point set
For any finite set \A of points in , we define a
-dimensional simple polyhedron whose face poset is isomorphic to the
poset of ``non-crossing marked graphs'' with vertex set \A, where a marked
graph is defined as a geometric graph together with a subset of its vertices.
The poset of non-crossing graphs on \A appears as the complement of the star
of a face in that polyhedron.
The polyhedron has a unique maximal bounded face, of dimension
where is the number of points of \A in the interior of \conv(\A). The
vertices of this polytope are all the pseudo-triangulations of \A, and the
edges are flips of two types: the traditional diagonal flips (in
pseudo-triangulations) and the removal or insertion of a single edge.
As a by-product of our construction we prove that all pseudo-triangulations
are infinitesimally rigid graphs.Comment: 28 pages, 16 figures. Main change from v1 and v2: Introduction has
been reshape
Lattice congruences, fans and Hopf algebras
We give a unified explanation of the geometric and algebraic properties of
two well-known maps, one from permutations to triangulations, and another from
permutations to subsets. Furthermore we give a broad generalization of the
maps. Specifically, for any lattice congruence of the weak order on a Coxeter
group we construct a complete fan of convex cones with strong properties
relative to the corresponding lattice quotient of the weak order. We show that
if a family of lattice congruences on the symmetric groups satisfies certain
compatibility conditions then the family defines a sub Hopf algebra of the
Malvenuto-Reutenauer Hopf algebra of permutations. Such a sub Hopf algebra has
a basis which is described by a type of pattern-avoidance. Applying these
results, we build the Malvenuto-Reutenauer algebra as the limit of an infinite
sequence of smaller algebras, where the second algebra in the sequence is the
Hopf algebra of non-commutative symmetric functions. We also associate both a
fan and a Hopf algebra to a set of permutations which appears to be
equinumerous with the Baxter permutations.Comment: 34 pages, 1 figur
Poset Ramsey number . III. N-shaped poset
Given partially ordered sets (posets) and , we
say that contains a copy of if for some injective function and for any , if and only if
. For any posets and , the poset Ramsey number
is the least positive integer such that no matter how the elements
of an -dimensional Boolean lattice are colored in blue and red, there is
either a copy of with all blue elements or a copy of with all red
elements.
We focus on the poset Ramsey number for a fixed poset and an
-dimensional Boolean lattice , as grows large. It is known that
, for positive constants and .
However, there is no poset known, for which , for
. This paper is devoted to a new method for finding upper bounds
on using a duality between copies of and sets of elements
that cover them, referred to as blockers. We prove several properties of
blockers and their direct relation to the Ramsey numbers. Using these
properties we show that , for a poset
with four elements and , such that , ,
, and the remaining pairs of elements are incomparable.Comment: 19 pages, 6 figure
- …