The class of Galled-Tree Explainable (GaTEx) graphs has just 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 of the tree correspond to the vertices of the graph. As a generalization, GaTEx graphs are precisely those graphs that can be uniquely represented by a particular rooted directed acyclic graph (called galled-tree). We consider here four prominent problems that are, in general, NP-hard: computing the size $\omega(G)$ of a maximum clique, the size $\chi(G)$ of an optimal vertex-coloring and the size $\alpha(G)$ of a maximum independent set of a given graph $G$ as well as determining whether a graph is perfectly orderable. We show here that $\omega(G)$, $\chi(G)$, $\alpha(G)$ can be computed in linear-time for GaTEx graphs $G$. The crucial idea for the linear-time algorithms is to avoid working on the GaTEx graphs $G$ directly, but to use the the galled-trees that explain $G$ as a guide for the algorithms to compute these invariants. In particular, we show first how to employ the galled-tree structure to compute a perfect ordering of GaTEx graphs in linear-time which is then used to determine $\omega(G)$, $\chi(G)$, $\alpha(G)$.

翻译：Galled-Tree可解释（GaTEx）图类最近被发现是余树（cographs）的自然推广。余树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，GaTEx图正是那些可由特定有根有向无环图（称为galled-tree）唯一表示的图。本文考虑四个在一般情况下为NP困难的经典问题：计算给定图$G$的最大团大小$\omega(G)$、最优顶点染色大小$\chi(G)$、最大独立集大小$\alpha(G)$，以及判断图是否为完美可排序图。我们证明，对于GaTEx图$G$，$\omega(G)$、$\chi(G)$、$\alpha(G)$可以在线性时间内计算。这一线性时间算法的关键思想在于不直接处理GaTEx图$G$本身，而是利用解释$G$的galled-tree结构作为算法计算这些不变量的引导。特别地，我们首先展示如何利用galled-tree结构在线性时间内为GaTEx图计算完美排序，进而利用该排序确定$\omega(G)$、$\chi(G)$、$\alpha(G)$。