  A dormant generic Miura $\mathfrak{sl}_2$-oper is a flat
$\mathrm{PGL}_2$-bundle over an algebraic curve in positive characteristic
equipped with some additional data. In the present paper, we give a
combinatorial description of dormant generic Miura $\mathfrak{sl}_2$-opers on a
totally degenerate curve. The combinatorial objects that we use are certain
branch numberings of $3$-regular graphs. Our description may be thought of as
an analogue of the combinatorial description of dormant $\mathfrak{sl}_2$-opers
given by S. Mochizuki, F. Liu, and B. Osserman. It allows us to think of the
Miura transformation in terms of combinatorics. As an application, we identify
the dormant generic Miura $\mathfrak{sl}_2$-opers on totally degenerate curves
of genus $>0$.
