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≥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.Comment: 33 page