The commuting graph of a group $G$ is the graph whose vertices are the elements of $G$, two distinct vertices joined if they commute. Our purpose in this paper is twofold: we discuss the computational problem of deciding whether a given graph is the commuting graph of a finite group; we give a quasipolynomial algorithm, and a polynomial algorithm for the case when the group is an extra\-special p-group for p an odd prime; we give new results on the question of whether the commuting graph of a given group is a cograph or a chordal graph, two classes of graphs defined by forbidden subgraphs. The problems are not unrelated, since there are a number of cases where hard computational problems on graphs are easier when restricted to special classes of graphs; we conjecture that the recognition problem is polynomial for cographs and chordal graphs.