767 research outputs found
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 matroids of branch-width three
For the abstract of this paper, please see the PDF file
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
Fork-decompositions of matroids
For the abstract of this paper, please see the PDF file
First order convergence of matroids
The model theory based notion of the first order convergence unifies the
notions of the left-convergence for dense structures and the Benjamini-Schramm
convergence for sparse structures. It is known that every first order
convergent sequence of graphs with bounded tree-depth can be represented by an
analytic limit object called a limit modeling. We establish the matroid
counterpart of this result: every first order convergent sequence of matroids
with bounded branch-depth representable over a fixed finite field has a limit
modeling, i.e., there exists an infinite matroid with the elements forming a
probability space that has asymptotically the same first order properties. We
show that neither of the bounded branch-depth assumption nor the
representability assumption can be removed.Comment: Accepted to the European Journal of Combinatoric
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: 36 pages, 2 figures. Final versio
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