The prime graph complements of ${\mathcal{A}}_{11}$- and ${\mathcal{A}}_{13}$-solvable groups

We study prime graph complements of ${\mathcal{A}}_n$-solvable groups, focusing on the cases ${\mathcal{A}}_{11}$ and ${\mathcal{A}}_{13}$. These provide the first classifications of $T$-solvable groups for which $\mathrm{\mid }\pi (T)\mathrm{\mid }>4$. We develop structural and representation-theoretic tools for analyzing the possible prime graph complements and use them to obtain complete characterizations for both ${\mathcal{A}}_{11}$-solvable and ${\mathcal{A}}_{13}$-solvable groups. The methods introduced also provide a framework for studying analogous classification problems for larger alternating groups.