  We investigate new graph classes of bounded mim-width, strictly extending
interval graphs and permutation graphs. The graphs $K_t \boxminus K_t$ and $K_t
\boxminus S_t$ are graphs obtained from the disjoint union of two cliques of
size $t$, and one clique of size $t$ and one independent set of size $t$
respectively, by adding a perfect matching. We prove that : (1) interval graphs
are $(K_3\boxminus S_3)$-free chordal graphs; and $(K_t\boxminus S_t)$-free
chordal graphs have mim-width at most $t-1$, (2) permutation graphs are
$(K_3\boxminus K_3)$-free co-comparability graphs; and $(K_t\boxminus
K_t)$-free co-comparability graphs have mim-width at most $t-1$, (3) chordal
graphs and co-comparability graphs have unbounded mim-width in general. We
obtain several algorithmic consequences; for instance, while Minimum Dominating
Set is NP-complete on chordal graphs, it can be solved in time
$n^{\mathcal{O}(t)}$ on $(K_t\boxminus S_t)$-free chordal graphs. The third
statement strengthens a result of Belmonte and Vatshelle stating that either
those classes do not have constant mim-width or a decomposition with constant
mim-width cannot be computed in polynomial time unless $P=NP$. We generalize
these ideas to bigger graph classes. We introduce a new width parameter
sim-width, of stronger modelling power than mim-width, by making a small change
in the definition of mim-width. We prove that chordal graphs and
co-comparability graphs have sim-width at most 1. We investigate a way to bound
mim-width for graphs of bounded sim-width by excluding $K_t\boxminus K_t$ and
$K_t\boxminus S_t$ as induced minors or induced subgraphs, and give algorithmic
consequences. Lastly, we show that circle graphs have unbounded sim-width, and
thus also unbounded mim-width.
