125 research outputs found
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
Complementation, Local Complementation, and Switching in Binary Matroids
In 2004, Ehrenfeucht, Harju, and Rozenberg showed that any graph on a vertex
set can be obtained from a complete graph on via a sequence of the
operations of complementation, switching edges and non-edges at a vertex, and
local complementation. The last operation involves taking the complement in the
neighbourhood of a vertex. In this paper, we consider natural generalizations
of these operations for binary matroids and explore their behaviour. We
characterize all binary matroids obtainable from the binary projective geometry
of rank under the operations of complementation and switching. Moreover, we
show that not all binary matroids of rank at most can be obtained from a
projective geometry of rank via a sequence of the three generalized
operations. We introduce a fourth operation and show that, with this additional
operation, we are able to obtain all binary matroids.Comment: Fixed an error in the proof of Theorem 5.3. Adv. in Appl. Math.
(2020
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
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
Linear rank-width of distance-hereditary graphs I. A polynomial-time algorithm
Linear rank-width is a linearized variation of rank-width, and it is deeply
related to matroid path-width. In this paper, we show that the linear
rank-width of every -vertex distance-hereditary graph, equivalently a graph
of rank-width at most , can be computed in time , and a linear layout witnessing the linear rank-width can be computed with
the same time complexity. As a corollary, we show that the path-width of every
-element matroid of branch-width at most can be computed in time
, provided that the matroid is given by an
independent set oracle.
To establish this result, we present a characterization of the linear
rank-width of distance-hereditary graphs in terms of their canonical split
decompositions. This characterization is similar to the known characterization
of the path-width of forests given by Ellis, Sudborough, and Turner [The vertex
separation and search number of a graph. Inf. Comput., 113(1):50--79, 1994].
However, different from forests, it is non-trivial to relate substructures of
the canonical split decomposition of a graph with some substructures of the
given graph. We introduce a notion of `limbs' of canonical split
decompositions, which correspond to certain vertex-minors of the original
graph, for the right characterization.Comment: 28 pages, 3 figures, 2 table. A preliminary version appeared in the
proceedings of WG'1
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
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