  A large number of NP-hard graph problems become polynomial-time solvable on
graph classes where the mim-width is bounded and quickly computable. Hence,
when solving such problems on special graph classes, it is helpful to know
whether the graph class under consideration has bounded mim-width. We first
extend the toolkit for proving (un)boundedness of mim-width of graph classes.
This enables us to initiate a systematic study into bounding mim-width from the
perspective of hereditary graph classes. For a given graph $H$, the class of
$H$-free graphs has bounded mim-width if and only if it has bounded
clique-width. We show that the same is not true for $(H_1,H_2)$-free graphs. We
find several general classes of $(H_1,H_2)$-free graphs having unbounded
clique-width, but the mim-width is bounded and quickly computable. We also
prove a number of new results showing that, for certain $H_1$ and $H_2$, the
class of $(H_1,H_2)$-free graphs has unbounded mim-width. Combining these with
known results, we present summary theorems of the current state of the art for
the boundedness of mim-width for $(H_1,H_2)$-free graphs.
