90 research outputs found
Evaluating the rank generating function of a graphic 2-polymatroid
We consider the complexity of the two-variable rank generating function, , of a graphic 2-polymatroid. For a graph , is the generating function for the number of subsets of edges of having a particular size and incident with a particular number of vertices of . We show that for any with , it is P-hard to evaluate at . We also consider the -thickening of a graph and computing for the -thickening of a graph
A deletion-contraction formula and monotonicity properties for the polymatroid Tutte polynomial
The Tutte polynomial is a crucial invariant of matroids. The polymatroid
Tutte polynomial , introduced by Bernardi et al., is an
extension of the classical Tutte polynomial from matroids to polymatroids .
In this paper, we first obtain a deletion-contraction formula for
. Then we prove two natural monotonicity properties, for
containment and for minors of the interior polynomial
and the exterior polynomial
, for polymatroids over . We show by a
counter-example that these monotonicity properties do not extend to
. Using deletion-contraction, we obtain formulas for the
coefficients of terms of degree in . Finally, for
all , we characterize hypergraphs so that the
coefficient of in the exterior polynomial of the associated polymatroid
attains its maximal value .Comment: 27 pages, 2 figure
Enumeration of -Polymatroids on up to Seven Elements
A theory of single-element extensions of integer polymatroids analogous to
that of matroids is developed. We present an algorithm to generate a catalog of
-polymatroids, up to isomorphism. When we implemented this algorithm on a
computer, obtaining all -polymatroids on at most seven elements, we
discovered the surprising fact that the number of -polymatroids on seven
elements fails to be unimodal in rank.Comment: 9 page
A version of Tutte's polynomial for hypergraphs
Tutte's dichromate T(x,y) is a well known graph invariant. Using the original
definition in terms of internal and external activities as our point of
departure, we generalize the valuations T(x,1) and T(1,y) to hypergraphs. In
the definition, we associate activities to hypertrees, which are
generalizations of the indicator function of the edge set of a spanning tree.
We prove that hypertrees form a lattice polytope which is the set of bases in a
polymatroid. In fact, we extend our invariants to integer polymatroids as well.
We also examine hypergraphs that can be represented by planar bipartite graphs,
write their hypertree polytopes in the form of a determinant, and prove a
duality property that leads to an extension of Tutte's Tree Trinity Theorem.Comment: 49 page
A splitter theorem for 3-connected 2-polymatroids
Seymour’s Splitter Theorem is a basic inductive tool for dealing with 3-connected matroids. This paper proves a generalization of that theorem for the class of 2-polymatroids. Such structures include matroids, and they model both sets of points and lines in a projective space and sets of edges in a graph. A series compression in such a structure is an analogue of contracting an edge of a graph that is in a series pair. A 2-polymatroid N is an s-minor of a 2-polymatroid M if N can be obtained from M by a sequence of contractions, series compressions, and dual-contractions, where the last are modified deletions. The main result proves that if M and N are 3-connected 2-polymatroids such that N is an s-minor of M, then M has a 3-connected s-minor M′ that has an s-minor isomorphic to N and has |E(M)| − 1 elements unless M is a whirl or the cycle matroid of a wheel. In the exceptional case, such an M′ can be found with |E(M)| − 2 elements
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