The class of $\mathsf{Ga}$lled-$\mathsf{T}$ree $\mathsf{Ex}$plainable ($\mathsf{GaTEx}$) graphs has recently been discovered as a natural generalization of cographs. Cographs are precisely those graphs that can be uniquely represented by a rooted tree where the leaves correspond to the vertices of the graph. As a generalization, $\mathsf{GaTEx}$ graphs are precisely those that can be uniquely represented by a particular rooted acyclic network, called a galled-tree. This paper explores the use of galled-trees to solve combinatorial problems on $\mathsf{GaTEx}$ graphs that are, in general, NP-hard. We demonstrate that finding a maximum clique, an optimal vertex coloring, a perfect order, as well as a maximum independent set in $\mathsf{GaTEx}$ graphs can be efficiently done in linear time. The key idea behind the linear-time algorithms is to utilize the galled-trees that explain the $\mathsf{GaTEx}$ graphs as a guide for computing the respective cliques, colorings, perfect orders, or independent sets.

翻译：$\mathsf{Ga}$lled-$\mathsf{T}$ree $\mathsf{Ex}$plainable ($\mathsf{GaTEx}$)图类最近被发现是余图的自然推广。余图正是那些可由有根树唯一表示的图，其中树的叶子对应图的顶点。作为推广，$\mathsf{GaTEx}$图则是由一种称为galled-tree的特定有根无环网络唯一表示的图。本文探索利用galled-tree求解$\mathsf{GaTEx}$图上一般NP难的组合问题。我们证明，在$\mathsf{GaTEx}$图中寻找最大团、最优顶点着色、完美序以及最大独立集均可在线性时间内高效完成。这些线性时间算法的核心思想是：利用解释$\mathsf{GaTEx}$图的galled-tree作为指导，分别计算相应的团、着色、完美序或独立集。