461 research outputs found
Algebraic matroids with graph symmetry
This paper studies the properties of two kinds of matroids: (a) algebraic
matroids and (b) finite and infinite matroids whose ground set have some
canonical symmetry, for example row and column symmetry and transposition
symmetry.
For (a) algebraic matroids, we expose cryptomorphisms making them accessible
to techniques from commutative algebra. This allows us to introduce for each
circuit in an algebraic matroid an invariant called circuit polynomial,
generalizing the minimal poly- nomial in classical Galois theory, and studying
the matroid structure with multivariate methods.
For (b) matroids with symmetries we introduce combinatorial invariants
capturing structural properties of the rank function and its limit behavior,
and obtain proofs which are purely combinatorial and do not assume algebraicity
of the matroid; these imply and generalize known results in some specific cases
where the matroid is also algebraic. These results are motivated by, and
readily applicable to framework rigidity, low-rank matrix completion and
determinantal varieties, which lie in the intersection of (a) and (b) where
additional results can be derived. We study the corresponding matroids and
their associated invariants, and for selected cases, we characterize the
matroidal structure and the circuit polynomials completely
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
Graph Theory
Highlights of this workshop on structural graph theory included new developments on graph and matroid minors, continuous structures arising as limits of finite graphs, and new approaches to higher graph connectivity via tree structures
Hypercellular graphs: partial cubes without as partial cube minor
We investigate the structure of isometric subgraphs of hypercubes (i.e.,
partial cubes) which do not contain finite convex subgraphs contractible to the
3-cube minus one vertex (here contraction means contracting the edges
corresponding to the same coordinate of the hypercube). Extending similar
results for median and cellular graphs, we show that the convex hull of an
isometric cycle of such a graph is gated and isomorphic to the Cartesian
product of edges and even cycles. Furthermore, we show that our graphs are
exactly the class of partial cubes in which any finite convex subgraph can be
obtained from the Cartesian products of edges and even cycles via successive
gated amalgams. This decomposition result enables us to establish a variety of
results. In particular, it yields that our class of graphs generalizes median
and cellular graphs, which motivates naming our graphs hypercellular.
Furthermore, we show that hypercellular graphs are tope graphs of zonotopal
complexes of oriented matroids. Finally, we characterize hypercellular graphs
as being median-cell -- a property naturally generalizing the notion of median
graphs.Comment: 35 pages, 6 figures, added example answering Question 1 from earlier
draft (Figure 6.
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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