140 research outputs found
Even-cycle decompositions of graphs with no odd--minor
An even-cycle decomposition of a graph G is a partition of E(G) into cycles
of even length. Evidently, every Eulerian bipartite graph has an even-cycle
decomposition. Seymour (1981) proved that every 2-connected loopless Eulerian
planar graph with an even number of edges also admits an even-cycle
decomposition. Later, Zhang (1994) generalized this to graphs with no
-minor.
Our main theorem gives sufficient conditions for the existence of even-cycle
decompositions of graphs in the absence of odd minors. Namely, we prove that
every 2-connected loopless Eulerian odd--minor-free graph with an even
number of edges has an even-cycle decomposition.
This is best possible in the sense that `odd--minor-free' cannot be
replaced with `odd--minor-free.' The main technical ingredient is a
structural characterization of the class of odd--minor-free graphs, which
is due to Lov\'asz, Seymour, Schrijver, and Truemper.Comment: 17 pages, 6 figures; minor revisio
An obstacle to a decomposition theorem for near-regular matroids
Seymour's Decomposition Theorem for regular matroids states that any matroid
representable over both GF(2) and GF(3) can be obtained from matroids that are
graphic, cographic, or isomorphic to R10 by 1-, 2-, and 3-sums. It is hoped
that similar characterizations hold for other classes of matroids, notably for
the class of near-regular matroids. Suppose that all near-regular matroids can
be obtained from matroids that belong to a few basic classes through k-sums.
Also suppose that these basic classes are such that, whenever a class contains
all graphic matroids, it does not contain all cographic matroids. We show that
in that case 3-sums will not suffice.Comment: 11 pages, 1 figur
The world of hereditary graph classes viewed through Truemper configurations
In 1982 Truemper gave a theorem that characterizes graphs whose edges can be labeled so that all chordless cycles have prescribed parities. The characterization states that this can be done for a graph G if and only if it can be done for all induced subgraphs of G that are of few speci c types, that we will call Truemper con gurations. Truemper was originally motivated by the problem of obtaining a co-NP characterization of bipartite graphs that are signable to be balanced (i.e. bipartite graphs whose node-node incidence matrices are balanceable matrices). The con gurations that Truemper identi ed in his theorem ended up playing a key role in understanding the structure of several seemingly diverse classes of objects, such as regular matroids, balanceable matrices and perfect graphs. In this survey we view all these classes, and more, through the excluded Truemper con gurations, focusing on the algorithmic consequences, trying to understand what structurally enables e cient recognition and optimization algorithms
Confinement of matroid representations to subsets of partial fields
Let M be a matroid representable over a (partial) field P and B a matrix
representable over a sub-partial field P' of P. We say that B confines M to P'
if, whenever a P-representation matrix A of M has a submatrix B, A is a scaled
P'-matrix. We show that, under some conditions on the partial fields, on M, and
on B, verifying whether B confines M to P' amounts to a finite check. A
corollary of this result is Whittle's Stabilizer Theorem.
A combination of the Confinement Theorem and the Lift Theorem from
arXiv:0804.3263 leads to a short proof of Whittle's characterization of the
matroids representable over GF(3) and other fields.
We also use a combination of the Confinement Theorem and the Lift Theorem to
prove a characterization, in terms of representability over partial fields, of
the 3-connected matroids that have k inequivalent representations over GF(5),
for k = 1, ..., 6.
Additionally we give, for a fixed matroid M, an algebraic construction of a
partial field P_M and a representation A over P_M such that every
representation of M over a partial field P is equal to f(A) for some
homomorphism f:P_M->P. Using the Confinement Theorem we prove an algebraic
analog of the theory of free expansions by Geelen et al.Comment: 45 page
The excluded minors for the class of matroids that are binary or ternary
We show that the excluded minors for the class of matroids that are binary or ternary are U2,5, U3,5, U2,4position indicatorF7, U2,4position indicatorF7*, U2,4position indicator2F7, U2,4position indicator2F7*, and the unique matroids obtained by relaxing a circuit-hyperplane in either AG(3,2) or T12. The proof makes essential use of results obtained by Truemper on the structure of almost-regular matroids. © 2011 Elsevier Ltd
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