59 research outputs found
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
Tangle-tree duality in abstract separation systems
We prove a general width duality theorem for combinatorial structures with
well-defined notions of cohesion and separation. These might be graphs and
matroids, but can be much more general or quite different. The theorem asserts
a duality between the existence of high cohesiveness somewhere local and a
global overall tree structure.
We describe cohesive substructures in a unified way in the format of tangles:
as orientations of low-order separations satisfying certain consistency axioms.
These axioms can be expressed without reference to the underlying structure,
such as a graph or matroid, but just in terms of the poset of the separations
themselves. This makes it possible to identify tangles, and apply our
tangle-tree duality theorem, in very diverse settings.
Our result implies all the classical duality theorems for width parameters in
graph minor theory, such as path-width, tree-width, branch-width or rank-width.
It yields new, tangle-type, duality theorems for tree-width and path-width. It
implies the existence of width parameters dual to cohesive substructures such
as -blocks, edge-tangles, or given subsets of tangles, for which no width
duality theorems were previously known.
Abstract separation systems can be found also in structures quite unlike
graphs and matroids. For example, our theorem can be applied to image analysis
by capturing the regions of an image as tangles of separations defined as
natural partitions of its set of pixels. It can be applied in big data contexts
by capturing clusters as tangles. It can be applied in the social sciences,
e.g. by capturing as tangles the few typical mindsets of individuals found by a
survey. It could also be applied in pure mathematics, e.g. to separations of
compact manifolds.Comment: We have expanded Section 2 on terminology for better readability,
adding explanatory text, examples, and figures. This paper replaces the first
half of our earlier paper arXiv:1406.379
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
The adjacency matroid of a graph
If is a looped graph, then its adjacency matrix represents a binary
matroid on . may be obtained from the delta-matroid
represented by the adjacency matrix of , but is less sensitive to
the structure of . Jaeger proved that every binary matroid is for
some [Ann. Discrete Math. 17 (1983), 371-376].
The relationship between the matroidal structure of and the
graphical structure of has many interesting features. For instance, the
matroid minors and are both of the form
where may be obtained from using local
complementation. In addition, matroidal considerations lead to a principal
vertex tripartition, distinct from the principal edge tripartition of
Rosenstiehl and Read [Ann. Discrete Math. 3 (1978), 195-226]. Several of these
results are given two very different proofs, the first involving linear algebra
and the second involving set systems or delta-matroids. Also, the Tutte
polynomials of the adjacency matroids of and its full subgraphs are closely
connected to the interlace polynomial of Arratia, Bollob\'{a}s and Sorkin
[Combinatorica 24 (2004), 567-584].Comment: v1: 19 pages, 1 figure. v2: 20 pages, 1 figure. v3:29 pages, no
figures. v3 includes an account of the relationship between the adjacency
matroid of a graph and the delta-matroid of a graph. v4: 30 pages, 1 figure.
v5: 31 pages, 1 figure. v6: 38 pages, 3 figures. v6 includes a discussion of
the duality between graphic matroids and adjacency matroids of looped circle
graph
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