613 research outputs found
The structure of 2-separations of infinite matroids
Generalizing a well known theorem for finite matroids, we prove that for
every (infinite) connected matroid M there is a unique tree T such that the
nodes of T correspond to minors of M that are either 3-connected or circuits or
cocircuits, and the edges of T correspond to certain nested 2-separations of M.
These decompositions are invariant under duality.Comment: 31 page
When can splits be drawn in the plane?
Split networks are a popular tool for the analysis and visualization of
complex evolutionary histories. Every collection of splits (bipartitions) of a
finite set can be represented by a split network. Here we characterize which
collection of splits can be represented using a planar split network. Our main
theorem links these collections of splits with oriented matroids and
arrangements of lines separating points in the plane. As a consequence of our
main theorem, we establish a particularly simple characterization of maximal
collections of these splits
Biconed graphs, edge-rooted forests, and h-vectors of matroid complexes
A well-known conjecture of Richard Stanley posits that the -vector of the
independence complex of a matroid is a pure -sequence. The
conjecture has been established for various classes but is open for graphic
matroids. A biconed graph is a graph with two specified `coning vertices', such
that every vertex of the graph is connected to at least one coning vertex. The
class of biconed graphs includes coned graphs, Ferrers graphs, and complete
multipartite graphs. We study the -vectors of graphic matroids arising from
biconed graphs, providing a combinatorial interpretation of their entries in
terms of `edge-rooted forests' of the underlying graph. This generalizes
constructions of Kook and Lee who studied the M\"obius coinvariant (the last
nonzero entry of the -vector) of graphic matroids of complete bipartite
graphs. We show that allowing for partially edge-rooted forests gives rise to a
pure multicomplex whose face count recovers the -vector, establishing
Stanley's conjecture for this class of matroids.Comment: 15 pages, 3 figures; V2: added omitted author to metadat
Rooted Cycle Bases
A cycle basis in an undirected graph is a minimal set of simple cycles whose
symmetric differences include all Eulerian subgraphs of the given graph. We
define a rooted cycle basis to be a cycle basis in which all cycles contain a
specified root edge, and we investigate the algorithmic problem of constructing
rooted cycle bases. We show that a given graph has a rooted cycle basis if and
only if the root edge belongs to its 2-core and the 2-core is
2-vertex-connected, and that constructing such a basis can be performed
efficiently. We show that in an unweighted or positively weighted graph, it is
possible to find the minimum weight rooted cycle basis in polynomial time.
Additionally, we show that it is NP-complete to find a fundamental rooted cycle
basis (a rooted cycle basis in which each cycle is formed by combining paths in
a fixed spanning tree with a single additional edge) but that the problem can
be solved by a fixed-parameter-tractable algorithm when parameterized by
clique-width.Comment: 12 pages with 10 additional pages of appendices and 10 figures.
Extended version of a paper to appear at the 14th Algorithms and Data
Structures Symposium (WADS), Victoria, BC, August 201
Topological representations of matroids
There is a one-to-one correspondence between geometric lattices and the
intersection lattices of arrangements of homotopy spheres. When the
arrangements are essential and fully partitioned, Zaslavsky's enumeration of
the cells of the arrangement still holds. An application of the theory shows
that all minimal cellular resolutions of matroid Steiner ideals are bounded
subcomplexes of homotopy sphere arrangements of the given matroid. As a result
the Betti numbers of the ideal are computed and seen to be equivalent to
Stanley's formula in the special case of face ideals of independence complexes
of matroids.Comment: 21 pages, 4 figures Numerous typographical and notational errors
corrected. Theorem 7.2 and the following discussion correcte
A Simplicial Tutte "5"-flow Conjecture
This paper concerns a generalization of nowhere-zero modular q-flows from
graphs to simplicial complexes of dimension d greater than 1. A modular q-flow
of a simplicial complex is an element of the kernel of the d-th boundary map
with coefficients in Z/qZ; it is called nowhere-zero if it is not zero
restricted to any of the facets of the complex. Briefly noting connections to
other invariants of simplicial complexes, this paper provides a generalization
of Tutte's 5-flow conjecture, which claims the universal existence of a 5-flow
for all bridgeless graphs. Once phrased, this paper concludes with bounds on
what "5" ought to be for simplicial complexes of dimension d: proving a lower
bound linear in d and a partial upper bound exponential in d.Comment: 26 pages. This is a late draft, soon to be submitted to a journal.
This paper has been presented at the Undergraduate Math Symposium at UI
On excluded minors for classes of graphical matroids
Frame matroids and lifted-graphic matroids are two distinct minor-closed
classes of matroids, each of which generalises the class of graphic matroids.
The class of quasi-graphic matroids, recently introduced by Geelen, Gerards,
and Whittle, simultaneously generalises both the classes of frame and
lifted-graphic matroids. Let be one of these three classes, and
let be a positive integer. We show that has only a finite
number of excluded minors of rank .Comment: 25 page
On the Complexity of Matroid Isomorphism Problem
We study the complexity of testing if two given matroids are isomorphic. The
problem is easily seen to be in . In the case of linear matroids,
which are represented over polynomially growing fields, we note that the
problem is unlikely to be -complete and is \co\NP-hard. We show
that when the rank of the matroid is bounded by a constant, linear matroid
isomorphism, and matroid isomorphism are both polynomial time many-one
equivalent to graph isomorphism. We give a polynomial time Turing reduction
from graphic matroid isomorphism problem to the graph isomorphism problem.
Using this, we are able to show that graphic matroid isomorphism testing for
planar graphs can be done in deterministic polynomial time. We then give a
polynomial time many-one reduction from bounded rank matroid isomorphism
problem to graphic matroid isomorphism, thus showing that all the above
problems are polynomial time equivalent. Further, for linear and graphic
matroids, we prove that the automorphism problem is polynomial time equivalent
to the corresponding isomorphism problems. In addition, we give a polynomial
time membership test algorithm for the automorphism group of a graphic matroid
Laminar Matroids
A laminar family is a collection of subsets of a set such
that, for any two intersecting sets, one is contained in the other. For a
capacity function on , let be \{I:|I\cap A|
\leq c(A)\text{ for all A\in\mathscr{A}}\}. Then is the
collection of independent sets of a (laminar) matroid on . We present a
method of compacting laminar presentations, characterize the class of laminar
matroids by their excluded minors, present a way to construct all laminar
matroids using basic operations, and compare the class of laminar matroids to
other well-known classes of matroids.Comment: 17 page
Spanning Circuits in Regular Matroids
We consider the fundamental Matroid Theory problem of finding a circuit in a
matroid spanning a set T of given terminal elements. For graphic matroids this
corresponds to the problem of finding a simple cycle passing through a set of
given terminal edges in a graph. The algorithmic study of the problem on
regular matroids, a superclass of graphic matroids, was initiated by
Gaven\v{c}iak, Kr\'al', and Oum [ICALP'12], who proved that the case of the
problem with |T|=2 is fixed-parameter tractable (FPT) when parameterized by the
length of the circuit. We extend the result of Gaven\v{c}iak, Kr\'al', and Oum
by showing that for regular matroids
- the Minimum Spanning Circuit problem, deciding whether there is a circuit
with at most \ell elements containing T, is FPT parameterized by k=\ell-|T|;
- the Spanning Circuit problem, deciding whether there is a circuit
containing T, is FPT parameterized by |T|. We note that extending our
algorithmic findings to binary matroids, a superclass of regular matroids, is
highly unlikely: Minimum Spanning Circuit parameterized by \ell is W[1]-hard on
binary matroids even when |T|=1. We also show a limit to how far our results
can be strengthened by considering a smaller parameter. More precisely, we
prove that Minimum Spanning Circuit parameterized by |T| is W[1]-hard even on
cographic matroids, a proper subclass of regular matroids
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