  We study a class of discrete memoryless broadcast interference channels
(DM-BICs), where one of the broadcast receivers is subject to the interference
from a point-to-point transmission. A general achievable rate region
$\mathcal{R}$ based on rate splitting, superposition coding and binning at the
broadcast transmitter and rate splitting at the interfering transmitter is
derived. Under two partial order broadcast conditions {\em
interference-oblivious less noisy} and {\em interference-cognizant less noisy},
a reduced form of $\mathcal{R}$ is shown to be equivalent to the region based
on a simpler scheme that uses only superposition coding at the broadcast
transmitter. Furthermore, the capacity regions of DM-BIC under the two partial
order broadcast conditions are characterized respectively for the strong and
very strong interference conditions.
