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 find characterizations in terms of minimal forbidden induced subgraphs for disconnected graphs, which imply the same for 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; in $\mathcal{O}(n^4)$-time for triangle-free graphs; and in $\mathcal{O}(n^2)$-time for graphs with bounded clique-width. We provide efficient algorithms to calculate the clique, the independence, the chromatic, and the treewidth numbers, as well as a minimum fill-in for Maya-Tupi graphs.