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.

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图表中的优先细度

区间图和适当区间图是非常有名的图类，相关文献很多。因此，已经提出了区间图的一些泛化，其中通过以某种特殊方式分割图，一般用 $k$ 区间图来表示图。作为这种方法的一个最近的例子，已经引入了 $k$-thin 和正确的 $k$-thin 图的类别，分别概括了区间和正确的区间图。即使对于固定的 $k\geq 2$，识别这些类别中的每一个的复杂性仍然是开放的。在这项工作中，我们引入了一个 $k$-thin 图（resp.proper $k$-thin graphs）的子类，称为 precedence $k$-thin graphs（resp.precedenceproper $k$-thin graphs）。关于分区优先级 $k$-thin 图形，我们提出了一种基于$PQ$-trees 的多项式时间识别算法。关于分区优先级正确的 $k$-thin 图，我们证明了相关的识别问题对于任意 $k$ 是 \NP 完全的，并且当 $k$ 固定时，多项式时间是可解的。此外，我们基于阈值图对这些类进行了表征。