475 research outputs found
Minimal classes of graphs of unbounded clique-width defined by finitely many forbidden induced subgraphs
We discover new hereditary classes of graphs that are minimal (with respect
to set inclusion) of unbounded clique-width. The new examples include split
permutation graphs and bichain graphs. Each of these classes is characterised
by a finite list of minimal forbidden induced subgraphs. These, therefore,
disprove a conjecture due to Daligault, Rao and Thomasse from 2010 claiming
that all such minimal classes must be defined by infinitely many forbidden
induced subgraphs.
In the same paper, Daligault, Rao and Thomasse make another conjecture that
every hereditary class of unbounded clique-width must contain a labelled
infinite antichain. We show that the two example classes we consider here
satisfy this conjecture. Indeed, they each contain a canonical labelled
infinite antichain, which leads us to propose a stronger conjecture: that every
hereditary class of graphs that is minimal of unbounded clique-width contains a
canonical labelled infinite antichain.Comment: 17 pages, 7 figure
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
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Minimal classes of graphs of unbounded clique-width defined by finitely many forbidden induced subgraphs
We discover new hereditary classes of graphs that are minimal (with respect to set inclusion) of unbounded clique-width. The new examples include split permutation graphs and bichain graphs. Each of these classes is characterised by a finite list of minimal forbidden induced subgraphs. These, therefore, disprove a conjecture due to Daligault, Rao and Thomassé from 2010 claiming that all such minimal classes must be defined by infinitely many forbidden induced subgraphs.
In the same paper, Daligault, Rao and Thomassé make another conjecture that every hereditary class of unbounded clique-width must contain a labelled infinite antichain. We show that the two example classes we consider here satisfy this conjecture. Indeed, they each contain a canonical labelled infinite antichain, which leads us to propose a stronger conjecture: that every hereditary class of graphs that is minimal of unbounded clique-width contains a canonical labelled infinite antichain
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
Outer Billiards on Kites
Outer billiards is a simple dynamical system based on a convex planar shape.
The Moser-Neumann question, first posed by B.H. Neumann around 1960, asks if
there exists a planar shape for which outer billiards has an unbounded orbit.
The first half of this monograph proves that outer billiards has an unbounded
orbit defined relative to any irrational kite. The second half of the monograph
gives a very sharp description of the set of unbounded orbits, both in terms of
the dynamics and the Hausdorff dimension. The analysis in both halves reveals a
close connection between outer billiards on kites and the modular group, as
well as connections to self-similar tilings, polytope exchange maps,
Diophantine approximation, and odometers.Comment: 296 pages. Essentially, I have added a "second half" to the previous
monograph. Parts I-IV are essentially the same as last posted version. Parts
V-VI have the new materia
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