Apex Graphs and Cographs

Authors

Jagdeep Singh, Binghamton University -- SUNYFollow
Vaidy Sivaraman, Mississippi State UniversityFollow
Thomas Zaslavsky, Binghamton University--SUNYFollow

Publication Date

December 2024

Abstract

A class G of graphs is called hereditary if it is closed under taking induced subgraphs. We denote by G^{apex} the class of graphs G that contain a vertex v such that G − v is in G. Borowiecki, Drgas-Burchardt, and Sidorowicz proved that if a hereditary class G has finitely many forbidden induced subgraphs, then so does G^{apex}. We provide an elementary proof of this result.

The hereditary class of cographs consists of all graphs G that can be generated from K_1 using complementation and disjoint union. A graph is an apex cograph if it contains a vertex whose deletion results in a cograph. Cographs are precisely the graphs that do not have the 4-vertex path as an induced subgraph. Our main result finds all such forbidden induced subgraphs for the class of apex cographs.

Creative Commons License

This work is licensed under a Creative Commons Attribution 4.0 License.

Recommended Citation

Singh, Jagdeep; Sivaraman, Vaidy; and Zaslavsky, Thomas (2024) "Apex Graphs and Cographs," Theory and Applications of Graphs: Vol. 11: Iss. 1, Article 4.
DOI: 10.20429/tag.2024.110104
Available at: https://digitalcommons.georgiasouthern.edu/tag/vol11/iss1/4