Maya-Tupi graphs: a generalization of split graphs

TL;DR

This paper introduces Maya-Tupi graphs, a new graph class generalizing split graphs, with characterizations and recognition algorithms for specific subclasses and graph families, expanding understanding of their structural properties.

Contribution

The paper defines Maya-Tupi graphs, characterizes them within certain classes, and develops efficient recognition algorithms, advancing the study of this new graph family.

Findings

Characterizations for disconnected graphs, trees, and cographs.

Linear time recognition algorithms for these subclasses.

Recognition in $\mathcal{O}(n^3)$-time for specific graph classes.

Abstract

We define the family of Maya-Tupi graphs as those graphs that admit a partition $(A,B)$ of their vertex sets such that $A$ induces a complete multipartite graph where each part has size at most two, and $B$ induces a graph where every connected component is $K_1$ or $K_2$. The family of Maya-Tupi graphs is self complementary, generalizes split graphs, falls into the sparse-dense partitioning schema and is characterized by finitely many forbidden induced subgraphs. Unfortunately, our computational experiments show that the number of minimal forbidden induced subgraphs to characterize Maya-Tupi graphs is greater than 2000. In this work, we study Maya-Tupi graphs when restricted to some well-known graph classes. We find characterizations in terms of minimal forbidden induced subgraphs for disconnected graphs, trees and cographs; our results imply linear time certifying recognition algorithms for Maya-Tupi graphs within these classes. We also show that Maya-Tupi graphs can be recognized in $\mathcal{O}(n^3)$-time in $C_4$-free graphs and in graphs with bounded neighborhood diversity.