Solving NP-hard problems on GaTEx graphs: Linear-time algorithms for perfect orderings, cliques, colorings, and independent sets

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...

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Bibliographic Details
Published in:Theoretical computer science Vol. 1037; p. 115157
Main Authors: Hellmuth, Marc, Scholz, Guillaume E.
Format: Journal Article
Language:English
Published: Elsevier B.V 11.05.2025
Subjects:
Cograph
Galled-tree
Linear-time algorithms
Modular decomposition
NP-hard problems
Modular decomposition
Galled-tree
NP-hard problems
Linear-time algorithms
Cograph
ISSN:0304-3975, 1879-2294
Online Access:Get full text
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Summary: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.
ISSN:0304-3975
1879-2294
DOI:10.1016/j.tcs.2025.115157