Forbidden subgraphs in commuting graphs of finite groups

Given a finite group $G$, the commuting graph of $G$ is the simple graph whose vertex set is $G$, and two distinct vertices are adjacent if they commute. In this paper, we classify all finite groups whose commuting graph is split and threshold. We also investigate the structure of a group whose commuting graph is either a cograph or a chordal graph, and determine all non-abelian finite simple groups whose commuting graph is a cograph. The results partly answer a question by Peter J. Cameron established in 2022 in his survey paper on graphs defined on groups.