  We investigate a degree preserving variant of the $\Delta$-Y transformation
which replaces a triangle with a new 6-valent vertex which has double edges to
the vertices that had been in the triangle. This operation is relevant for
understanding scalar Feynman integrals in 6 dimensions. We study the structure
of equivalence classes under this operation and its inverse, with particular
attention to when the equivalence classes are finite, when they contain simple
6-regular graphs, and when they contain doubled 3-regular graphs. The last of
these, in particular, is relevant for the Feynman integral calculations and we
make some observations linking the structure of these classes to the Feynman
periods. Furthermore, we investigate properties of minimal graphs in these
equivalence classes.
