878 research outputs found
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
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
Branch-depth: Generalizing tree-depth of graphs
We present a concept called the branch-depth of a connectivity function, that
generalizes the tree-depth of graphs. Then we prove two theorems showing that
this concept aligns closely with the notions of tree-depth and shrub-depth of
graphs as follows. For a graph and a subset of we let
be the number of vertices incident with an edge in and an
edge in . For a subset of , let be the rank
of the adjacency matrix between and over the binary field.
We prove that a class of graphs has bounded tree-depth if and only if the
corresponding class of functions has bounded branch-depth and
similarly a class of graphs has bounded shrub-depth if and only if the
corresponding class of functions has bounded branch-depth, which we
call the rank-depth of graphs.
Furthermore we investigate various potential generalizations of tree-depth to
matroids and prove that matroids representable over a fixed finite field having
no large circuits are well-quasi-ordered by the restriction.Comment: 34 pages, 2 figure
On Feedback Vertex Set: New Measure and New Structures
We present a new parameterized algorithm for the {feedback vertex set}
problem ({\sc fvs}) on undirected graphs. We approach the problem by
considering a variation of it, the {disjoint feedback vertex set} problem ({\sc
disjoint-fvs}), which finds a feedback vertex set of size that has no
overlap with a given feedback vertex set of the graph . We develop an
improved kernelization algorithm for {\sc disjoint-fvs} and show that {\sc
disjoint-fvs} can be solved in polynomial time when all vertices in have degrees upper bounded by three. We then propose a new
branch-and-search process on {\sc disjoint-fvs}, and introduce a new
branch-and-search measure. The process effectively reduces a given graph to a
graph on which {\sc disjoint-fvs} becomes polynomial-time solvable, and the new
measure more accurately evaluates the efficiency of the process. These
algorithmic and combinatorial studies enable us to develop an
-time parameterized algorithm for the general {\sc fvs} problem,
improving all previous algorithms for the problem.Comment: Final version, to appear in Algorithmic
Mutant knots and intersection graphs
We prove that if a finite order knot invariant does not distinguish mutant
knots, then the corresponding weight system depends on the intersection graph
of a chord diagram rather than on the diagram itself. The converse statement is
easy and well known. We discuss relationship between our results and certain
Lie algebra weight systems.Comment: 13 pages, many figure
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
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