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. 
Structure of graphs that exclude a perfect matching on four vertices as an induced graph
graph theory
Yes; they are known as $2K_2$-free graphs. Some additional references:On Toughness and Hamiltonicity of $2K_2$-Free GraphsThe maximum number of edges in $2K_2$-free graphs of bounded degree;Two characterisations of minimal triangulations of $2_{K2}$-free graphs