2019-08-01 Sang-il Oum (엄상일), Branch depth: Generalizing tree depth of graphs

IBS Discrete Mathematics Group
IBS Summer Research Program on Algorithms and Complexity in Discrete Structures
Sang-il Oum (엄상일), Branch-depth: Generalizing tree-depth of graphs
August 01 2019, Thursday @ 11:30 AM ~ 12:00 PM
Room B232, IBS (기초과학연구원)
Speaker
Sang-il Oum (엄상일)
IBS Discrete Mathematics Group and KAIST, Korea
dimag.ibs.re.k...
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 $G=(V,E)$ and a subset $A$ of $E$ we let $\lambda_G(A)$ be the number of vertices incident with an edge in $A$ and an edge in $E∖A$. For a subset $X$ of $V$, let $
ho_G(X)$ be the rank of the adjacency matrix between $X$ and $V∖X$ over the binary field. We prove that a class of graphs has bounded tree-depth if and only if the corresponding class of functions $\lambda_G$ has bounded branch-depth and similarly a class of graphs has bounded shrub-depth if and only if the corresponding class of functions $
ho_G$ 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.
This is a joint work with Matt DeVos and O-joung Kwon.