Precedence thinness in graphs

Abstract

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.