  Interval and proper interval graphs are very well-known graph classes, for
which there is a wide literature. As a consequence, some generalizations of
interval graphs have been proposed, in which graphs in general are expressed in
terms of $k$ interval graphs, by splitting the graph in some special way.
  As a recent example of such an approach, the classes of $k$-thin and proper
$k$-thin graphs have been introduced generalizing interval and proper interval
graphs, respectively. The complexity of the recognition of each of these
classes is still open, even for fixed $k \geq 2$.
  In this work, we introduce a subclass of $k$-thin graphs (resp. proper
$k$-thin graphs), called precedence $k$-thin graphs (resp. precedence proper
$k$-thin graphs). Concerning partitioned precedence $k$-thin graphs, we present
a polynomial time recognition algorithm based on $PQ$-trees. With respect to
partitioned precedence proper $k$-thin graphs, we prove that the related
recognition problem is \NP-complete for an arbitrary $k$ and polynomial-time
solvable when $k$ is fixed. Moreover, we present a characterization for these
classes based on threshold graphs.
