The class of GAlled-Tree EXplainable (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, 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 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 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 GATEX graphs as a guide for computing the respective cliques, colorings, perfect orders, or independent sets.