Consider the minimal class of (simple, undirected) connected graphs (strictly speaking, isomorphism classes of connected graphs) which contains a single vertex $K_1$, and is closed under following operations:

add a copy $\tilde{v}$ of a vertex $v$ (neighbours of the new vertex are the same as of $v$) and do not join it with $v$ (the new graph must be connected, in other words, you are not allowed to start with this operation applied to $K_1$);

add a copy $\tilde{v}$ of a vertex $v$ (neighbours of the new vertex are the same as of $v$) and join it with $v$;

glue two graphs by a vertex.

Alternative description of this class (I hope that we have a proof, it is not very short) is the following: any simple cycle with at least 5 vertices must have two disjoint chords.

Was such class studied?