  A moplex is a natural graph structure that arises when lifting Dirac's
classical theorem from chordal graphs to general graphs. While every
non-complete graph has at least two moplexes, little is known about structural
properties of graphs with a bounded number of moplexes. The study of these
graphs is, in part, motivated by the parallel between moplexes in general
graphs and simplicial modules in chordal graphs: unlike in the moplex setting,
properties of chordal graphs with a bounded number of simplicial modules are
well understood. For instance, chordal graphs having at most two simplicial
modules are interval.
  In this work, we initiate an investigation of $k$-moplex graphs, which are
defined as graphs containing at most $k$ moplexes. Of particular interest is
the smallest nontrivial case $k=2$, which forms a counterpart to the class of
interval graphs. As our main structural result, we show that, when restricted
to connected graphs, the class of $2$-moplex graphs is sandwiched between the
classes of proper interval graphs and cocomparability graphs; moreover, both
inclusions are tight for hereditary classes. From a complexity theoretic
viewpoint, this leads to the natural question of whether the presence of at
most two moplexes guarantees a sufficient amount of structure to efficiently
solve problems that are known to be intractable on cocomparability graphs, but
not on proper interval graphs. We develop new reductions that answer this
question negatively for two prominent problems fitting this profile, namely
Graph Isomorphism and Max-Cut. On the other hand, we prove that every connected
$2$-moplex graph contains a Hamiltonian path, generalising the same property of
connected proper interval graphs.
