Precedence thinness in graphs

TL;DR

This paper introduces precedence $k$-thin graphs, a subclass of $k$-thin graphs, providing polynomial recognition algorithms for some cases and proving NP-completeness for others, advancing understanding of generalized interval graph classes.

Contribution

It defines precedence $k$-thin graphs, offers polynomial algorithms for their recognition in certain cases, and characterizes these classes using threshold graphs.

Findings

Polynomial recognition algorithm for partitioned precedence $k$-thin graphs.

NP-completeness of recognition for partitioned precedence proper $k$-thin graphs for arbitrary $k$.

Characterization of classes via threshold 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 $k \geq 2$. 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.