Precedence thinness in graphs

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