5 research outputs found
Obstructions for bounded shrub-depth and rank-depth
Shrub-depth and rank-depth are dense analogues of the tree-depth of a graph.
It is well known that a graph has large tree-depth if and only if it has a long
path as a subgraph. We prove an analogous statement for shrub-depth and
rank-depth, which was conjectured by Hlin\v{e}n\'y, Kwon, Obdr\v{z}\'alek, and
Ordyniak [Tree-depth and vertex-minors, European J.~Combin. 2016]. Namely, we
prove that a graph has large rank-depth if and only if it has a vertex-minor
isomorphic to a long path. This implies that for every integer , the class
of graphs with no vertex-minor isomorphic to the path on vertices has
bounded shrub-depth.Comment: 19 pages, 5 figures; accepted to Journal of Combinatorial Theory Ser.
Obstructions for bounded branch-depth in matroids
DeVos, Kwon, and Oum introduced the concept of branch-depth of matroids as a
natural analogue of tree-depth of graphs. They conjectured that a matroid of
sufficiently large branch-depth contains the uniform matroid or the
cycle matroid of a large fan graph as a minor. We prove that matroids with
sufficiently large branch-depth either contain the cycle matroid of a large fan
graph as a minor or have large branch-width. As a corollary, we prove their
conjecture for matroids representable over a fixed finite field and
quasi-graphic matroids, where the uniform matroid is not an option.Comment: 25 pages, 1 figur
Stable graphs of bounded twin-width
We prove that every class of graphs that is monadically stable
and has bounded twin-width can be transduced from some class with bounded
sparse twin-width. This generalizes analogous results for classes of bounded
linear cliquewidth and of bounded cliquewidth. It also implies that monadically
stable classes of bounded twin-widthare linearly -bounded.Comment: 44 pages, 2 figure