261 research outputs found
The Group Structure of Pivot and Loop Complementation on Graphs and Set Systems
We study the interplay between principal pivot transform (pivot) and loop
complementation for graphs. This is done by generalizing loop complementation
(in addition to pivot) to set systems. We show that the operations together,
when restricted to single vertices, form the permutation group S_3. This leads,
e.g., to a normal form for sequences of pivots and loop complementation on
graphs. The results have consequences for the operations of local
complementation and edge complementation on simple graphs: an alternative proof
of a classic result involving local and edge complementation is obtained, and
the effect of sequences of local complementations on simple graphs is
characterized.Comment: 21 pages, 7 figures, significant additions w.r.t. v3 are Thm 7 and
Remark 2
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
Interlace Polynomials for Multimatroids and Delta-Matroids
We provide a unified framework in which the interlace polynomial and several
related graph polynomials are defined more generally for multimatroids and
delta-matroids. Using combinatorial properties of multimatroids rather than
graph-theoretical arguments, we find that various known results about these
polynomials, including their recursive relations, are both more efficiently and
more generally obtained. In addition, we obtain several interrelationships and
results for polynomials on multimatroids and delta-matroids that correspond to
new interrelationships and results for the corresponding graphs polynomials. As
a tool we prove the equivalence of tight 3-matroids and delta-matroids closed
under the operations of twist and loop complementation, called vf-safe
delta-matroids. This result is of independent interest and related to the
equivalence between tight 2-matroids and even delta-matroids observed by
Bouchet.Comment: 35 pages, 3 figure
The Interlace Polynomial
In this paper, we survey results regarding the interlace polynomial of a
graph, connections to such graph polynomials as the Martin and Tutte
polynomials, and generalizations to the realms of isotropic systems and
delta-matroids.Comment: 18 pages, 5 figures, to appear as a chapter in: Graph Polynomials,
edited by M. Dehmer et al., CRC Press/Taylor & Francis Group, LL
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
Quaternary matroids are vf-safe
Binary delta-matroids are closed under vertex flips, which consist of the
natural operations of twist and loop complementation. In this note we provide
an extension of this result from GF(2) to GF(4). As a consequence, quaternary
matroids are "safe" under vertex flips (vf-safe for short). As an application,
we find that the matroid of a bicycle space of a quaternary matroid is
independent of the chosen representation. This extends a result of Vertigan [J.
Comb. Theory B (1998)] concerning the bicycle dimension of quaternary matroids.Comment: 8 pages, no figures, the contents of this paper is now merged into v2
of [arXiv:1210.7718] (except for this comment, v2 is identical to v1
Well-Quasi-Ordering of Matrices under Schur Complement and Applications to Directed Graphs
In [Rank-Width and Well-Quasi-Ordering of Skew-Symmetric or Symmetric
Matrices, arXiv:1007.3807v1] Oum proved that, for a fixed finite field
, any infinite sequence of (skew) symmetric matrices
over of bounded -rank-width has a pair , such
that is isomorphic to a principal submatrix of a principal pivot
transform of . We generalise this result to -symmetric matrices
introduced by Rao and myself in [The Rank-Width of Edge-Coloured Graphs,
arXiv:0709.1433v4]. (Skew) symmetric matrices are special cases of
-symmetric matrices. As a by-product, we obtain that for every infinite
sequence of directed graphs of bounded rank-width there exist a
pair such that is a pivot-minor of . Another consequence is
that non-singular principal submatrices of a -symmetric matrix form a
delta-matroid. We extend in this way the notion of representability of
delta-matroids by Bouchet.Comment: 35 pages. Revised version with a section for directed graph
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