I am interested in understanding the structure of the class of graphs $G$ such that there is no vertex induced subgraph on four vertices that is a perfect matching. Stated differently for any four vertices $a,b,c,d$ in $G$ if $ab$ and $cd$ are edges then the graph should have at least one more edge on the four vertices. Has this class been studied previously? Any references or insights would be appreciated. We understand this class when restricted to bipartite graphs but the general case seems more tricky.
Yes; they are known as $2K_2$-free graphs. Some additional references: