  An edge $e$ of a matching covered graph $G$ is removable if $G-e$ is also
matching covered. Carvalho, Lucchesi, and Murty showed that every brick $G$
different from $K_4$ and $\overline{C_6}$ has at least $\Delta-2$ removable
edges, where $\Delta$ is the maximum degree of $G$. In this paper, we
generalize the result to irreducible near-bricks, where a graph is irreducible
if it contains no single ear of length three or more.
