70 research outputs found
Valuative invariants for polymatroids
Many important invariants for matroids and polymatroids, such as the Tutte
polynomial, the Billera-Jia-Reiner quasi-symmetric function, and the invariant
introduced by the first author, are valuative. In this paper we
construct the -modules of all -valued valuative functions for labeled
matroids and polymatroids on a fixed ground set, and their unlabeled
counterparts, the -modules of valuative invariants. We give explicit bases
for these modules and for their dual modules generated by indicator functions
of polytopes, and explicit formulas for their ranks. Our results confirm a
conjecture of the first author that is universal for valuative
invariants.Comment: 54 pp, 9 figs. Mostly minor changes; Cor 10.5 and formula for
products of s corrected; Prop 7.2 is new. To appear in Advances in
Mathematic
Connectivity of Matroids and Polymatroids
This dissertation is a collection of work on matroid and polymatroid connectivity. Connectivity is a useful property of matroids that allows a matroid to be decomposed naturally into its connected components, which are like blocks in a graph. The Cunningham-Edmonds tree decomposition further gives a way to decompose matroids into 3-connected minors. Much of the research below concerns alternate senses in which matroids and polymatroids can be connected. After a brief introduction to matroid theory in Chapter 1, the main results of this dissertation are given in Chapters 2 and 3. Tutte proved that, for an element e of a 2- connected matroid M , either the deletion or the contraction of e for M is 2-connected. In Chapter 2, a new notion of matroid connectivity is defined and it is shown that this new notion only enjoys the above inductive property when it agrees with the usual notion of 2-connectivity. Another result is proved to reinforce the special importance of this usual notion. In Chapter 3, a result of Brylawski and Seymour is considered. That result extends Tutte’s theorem by showing that if the element e is chosen to avoid a 2-connected minor N of M, then the deletion or contraction of e form M is not only 2-connected but maintains N as a minor. The main result of Chapter 3 proves an analogue of this result for 2-polymatroids, a natural extension of matroids. Chapter 4 describes a class of binary matroids that generalizes cubic graphs. Specifically, attention is focused on binary matroids having a cocircuit basis where every cocircuit in the basis, as well as the symmetric difference of all these cocircuits, has precisely three elements
A note on the connectivity of 2-polymatroid minors
Brylawski and Seymour independently proved that if is a connected matroid
with a connected minor , and , then or
is connected having as a minor. This paper proves an analogous but
somewhat weaker result for -polymatroids. Specifically, if is a
connected -polymatroid with a proper connected minor , then there is an
element of such that or is connected
having as a minor. We also consider what can be said about the uniqueness
of the way in which the elements of can be removed so that
connectedness is always maintained.Comment: 9 page
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