Thinness and its variations on some graph families and coloring graphs of bounded thinness
Flavia Bonomo-Braberman, Eric Brandwein, Fabiano S. Oliveira, Moysés S. Sampaio, Agustin Sansone, Jayme L. Szwarcfiter
TL;DR
The paper investigates how bounded thinness generalizes interval graphs by studying (proper) $k$-thin graphs and several variants across crown graphs $CR_n$, cographs, and grids $GR_{n,m}$. It develops structural characterizations and constructive layouts to obtain exact thinness values for $CR_n$, polynomial-time computability for precedence thinness on cographs, and tighter bounds for grid thinness and precedence thinness, including the specialized case of $GR_{2,n}$. It further establishes the computational complexity of coloring on these classes, proving NP-completeness for precedence $2$-thin and proper $2$-thin graphs with $k$ part of the input, while showing polynomial-time solvability for precedence proper $2$-thin graphs due to perfect orderability. These results advance understanding of how thinness controls algorithmic feasibility in colored graph problems and offer practical partition/order constructions for important graph families.
Abstract
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.