  Treewidth is an important graph invariant, relevant for both structural and
algorithmic reasons. A necessary condition for a graph class to have bounded
treewidth is the absence of large cliques. We study graph classes closed under
taking induced subgraphs in which this condition is also sufficient, which we
call $(tw,\omega)$-bounded. Such graph classes are known to have useful
algorithmic applications related to variants of the clique and $k$-coloring
problems. We consider six well-known graph containment relations: the minor,
topological minor, subgraph, induced minor, induced topological minor, and
induced subgraph relations. For each of them, we give a complete
characterization of the graphs $H$ for which the class of graphs excluding $H$
is $(tw,\omega)$-bounded. Our results yield an infinite family of
$\chi$-bounded induced-minor-closed graph classes and imply that the class of
$1$-perfectly orientable graphs is $(tw,\omega)$-bounded, leading to
linear-time algorithms for $k$-coloring $1$-perfectly orientable graphs for
every fixed~$k$. This answers a question of Bre\v sar, Hartinger, Kos, and
Milani{\v c} from 2018 and one of Beisegel, Chudnovsky, Gurvich, Milani{\v c},
and Servatius from 2019, respectively. We also reveal some further algorithmic
implications of $(tw,\omega)$-boundedness related to list $k$-coloring and
clique problems. In addition, we propose a question about the complexity of the
maximum weight independent set problem in $(tw,\omega)$-bounded graph classes
and prove that the problem is polynomial-time solvable in every class of graphs
excluding a fixed star as an induced minor.
