Interval graphs and proper interval graphs are well known graph classes, for which several generalizations have been proposed in the literature. In this work, we study the (proper) thinness, and several variations, for the classes of cographs, crowns graphs and grid graphs. We provide the exact values for several variants of thinness (proper, independent, complete, precedence, and combinations of them) for the crown graphs $CR_n$. For cographs, we prove that the precedence thinness can be determined in polynomial time. We also improve known bounds for the thinness of $n \times n$ grids $GR_n$ and $m \times n$ grids $GR_{m,n}$, proving that $\left \lceil \frac{n-1}{3} \right \rceil \leq \mbox{thin}(GR_n) \leq \left \lceil \frac{n+1}{2} \right \rceil$. Regarding the precedence thinness, we prove that $\mbox{prec-thin}(GR_{n,2}) = \left \lceil \frac{n+1}{2} \right \rceil$ and that $\left \lceil \frac{n-1}{3} \right \rceil \left \lceil\frac{n-1}{2} \right \rceil + 1 \leq \mbox{prec-thin}(GR_n) \leq \left \lceil\frac{n-1}{2} \right \rceil^2+1$. As applications, we show that the $k$-coloring problem is NP-complete for precedence $2$-thin graphs and for proper $2$-thin graphs, when $k$ is part of the input. On the positive side, it is polynomially solvable for precedence proper $2$-thin graphs, given the order and partition.

翻译：区间图和真区间图是众所周知的图类，已有文献提出了它们的多种推广形式。本文研究了余图、冠图和网格图的（真）薄度及其多种变体。我们确定了冠图$CR_n$的若干薄度变体（真薄度、独立薄度、完全薄度、优先薄度及其组合）的精确值。对于余图，我们证明了优先薄度可在多项式时间内确定。此外，我们改进了$n \times n$网格图$GR_n$和$m \times n$网格图$GR_{m,n}$的薄度已知界，证明了$\left \lceil \frac{n-1}{3} \right \rceil \leq \mbox{thin}(GR_n) \leq \left \lceil \frac{n+1}{2} \right \rceil$。关于优先薄度，我们证明了$\mbox{prec-thin}(GR_{n,2}) = \left \lceil \frac{n+1}{2} \right \rceil$且$\left \lceil \frac{n-1}{3} \right \rceil \left \lceil\frac{n-1}{2} \right \rceil + 1 \leq \mbox{prec-thin}(GR_n) \leq \left \lceil\frac{n-1}{2} \right \rceil^2+1$。作为应用，我们证明了当$k$作为输入参数时，$k$着色问题在优先$2$-薄图和真$2$-薄图上均为NP完全问题。从积极方面看，给定顺序和划分后，该问题可在多项式时间内求解于优先真$2$-薄图。