• DOI:10.48550/arXiv.2211.16854
• Corpus ID: 254095925

Resolving Prime Modules: The Structure of Pseudo-cographs and Galled-Tree Explainable Graphs

@article{Hellmuth2022ResolvingPM,
  title={Resolving Prime Modules: The Structure of Pseudo-cographs and Galled-Tree Explainable Graphs},
  author={Marc Hellmuth and Guillaume E. Scholz},
  journal={ArXiv},
  year={2022},
  volume={abs/2211.16854}
}

• M. Hellmuth, G. Scholz
• Published 30 November 2022
• Mathematics
• ArXiv

The modular decomposition of a graph G is a natural construction to capture key features of G in terms of a labeled tree ( T , t ) whose vertices are labeled as “series” (1), “parallel” (0) or “prime”. However, full information of G is provided by its modular decomposition tree ( T , t ) only, if G is a cograph, i.e., G does not contain prime modules. In this case, ( T , t ) explains G , i.e., { x , y } ∈ E ( G ) if and only if the lowest common ancestor lca T ( x , y ) of x and y has label “1”. Pseudo-cographs, or, more general, G A TE X graphs G are graphs that can be explained by labeled galled-trees, i.e., labeled networks ( N , t ) that are obtained from the modular decomposition tree ( T , t ) of G by replacing the prime vertices in T by simple labeled cycles. G A TE X graphs can be recognized and labeled galled-trees that explain these graphs can be constructed in linear time. In this contribution, we provide a novel characterization of G A TE X graphs in terms of a set F GT of 25 forbidden induced subgraphs. This characterization, in turn, allows us to show that G A TE X graphs are closely related to many other well-known graph classes such as P 4 -sparse and P 4 -reducible graphs, weakly-chordal graphs, perfect graphs with perfect order, comparability and permutation graphs, murky graphs as well as interval graphs, Meyniel graphs or very strongly-perfect and brittle graphs.