How to detect if any of 2 columns in a Pandas data frame (with N columns) have a parent-child relationship?

Question

I need to efficiently detect if any pair of 2 columns in a data frame has a parent/child relationship. The purpose is to automatically detect all such relations in a data frame. The data frame is not known beforehand. Efficiently, because I already known that N*(N-1)*2 column-to-column investigations need to be executed. Below there is a sample dataset for this tricky problem. Ideally there would be a function that returns all column pairs that constitute a parent child relation, e.g.

resolve_parent_child_dependencies(df)

For the sample below (ignoring all the other columns) this should return

[(col1, col2), (col1, col3), (col1, col4),(col1, colN) (col2, col3), ..., (colN, col4)]

Here's the sample dataset.

col1 col2 col3 col4 ... colN
   A   A1  A11  foo        X        
   A   A2  A21  bar        Y
   A   A2  A22  foo        X
   B   B1  B11  baz        Z
   B   B2  B21  qux        Z
   B   B3  B22  baz        Z

Purpose is to turn any pandas data frame without much configuration into a multi-dimensional database (the one im currently developing, https://tinyolap.com). The function is needed to auto-detect all the relationships to derive the dimensional model from the data frame.

Any ideas to get me restarted? 100x thanks...