21.24 (2026)

For a finite group $G$, the power graph $\mathcal{P}(G)$ is the graph with vertex set $G$ and edges $\{x, y\}$ for all $x \neq y \in G$ such that either $x \in \langle y \rangle$ or $y \in \langle x \rangle$. Is it true that, for every finite group $G$, if $\mathcal{P}(G)$ is a cograph, then $\mathcal{P}(G)$ is chordal? Cf. 21.23.

This holds if every element of $G$ has prime power order (D. Bubboloni, F. Fumagalli, C. E. Praeger, Preprint, 2025, https://arxiv.org/abs/2510.18073) and if $G$ is a nonabelian simple group (J. Cameron, P. Manna, R. Mehatari, J. Algebra, 591 (2022), 59–74; J. Brachter, E. Kaja, J. Algebr. Comb., 58 (2023), 1095–1124).