391 research outputs found
Dessins, their delta-matroids and partial duals
Given a map on a connected and closed orientable surface, the
delta-matroid of is a combinatorial object associated to which captures some topological information of the embedding. We explore how
delta-matroids associated to dessins d'enfants behave under the action of the
absolute Galois group. Twists of delta-matroids are considered as well; they
correspond to the recently introduced operation of partial duality of maps.
Furthermore, we prove that every map has a partial dual defined over its field
of moduli. A relationship between dessins, partial duals and tropical curves
arising from the cartography groups of dessins is observed as well.Comment: 34 pages, 20 figures. Accepted for publication in the SIGMAP14
Conference Proceeding
Binary matroids and local complementation
We introduce a binary matroid M(IAS(G)) associated with a looped simple graph
G. M(IAS(G)) classifies G up to local equivalence, and determines the
delta-matroid and isotropic system associated with G. Moreover, a parametrized
form of its Tutte polynomial yields the interlace polynomials of G.Comment: This article supersedes arXiv:1301.0293. v2: 26 pages, 2 figures. v3
- v5: 31 pages, 2 figures v6: Final prepublication versio
Nullity and Loop Complementation for Delta-Matroids
We show that the symmetric difference distance measure for set systems, and
more specifically for delta-matroids, corresponds to the notion of nullity for
symmetric and skew-symmetric matrices. In particular, as graphs (i.e.,
symmetric matrices over GF(2)) may be seen as a special class of
delta-matroids, this distance measure generalizes the notion of nullity in this
case. We characterize delta-matroids in terms of equicardinality of minimal
sets with respect to inclusion (in addition we obtain similar characterizations
for matroids). In this way, we find that, e.g., the delta-matroids obtained
after loop complementation and after pivot on a single element together with
the original delta-matroid fulfill the property that two of them have equal
"null space" while the third has a larger dimension.Comment: Changes w.r.t. v4: different style, Section 8 is extended, and in
addition a few small changes are made in the rest of the paper. 15 pages, no
figure
Even Delta-Matroids and the Complexity of Planar Boolean CSPs
The main result of this paper is a generalization of the classical blossom
algorithm for finding perfect matchings. Our algorithm can efficiently solve
Boolean CSPs where each variable appears in exactly two constraints (we call it
edge CSP) and all constraints are even -matroid relations (represented
by lists of tuples). As a consequence of this, we settle the complexity
classification of planar Boolean CSPs started by Dvorak and Kupec.
Using a reduction to even -matroids, we then extend the tractability
result to larger classes of -matroids that we call efficiently
coverable. It properly includes classes that were known to be tractable before,
namely co-independent, compact, local, linear and binary, with the following
caveat: we represent -matroids by lists of tuples, while the last two
use a representation by matrices. Since an matrix can represent
exponentially many tuples, our tractability result is not strictly stronger
than the known algorithm for linear and binary -matroids.Comment: 33 pages, 9 figure
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