Structure of graphs that exclude a perfect matching on four vertices as an induced graph
Yes; they are known as $2K_2$-free graphs. Some additional references:


"On Toughness and Hamiltonicity of $2K_2$-Free Graphs"
"The maximum number of edges in $2K_2$-free graphs of bounded degree";
"Two characterisations of minimal triangulations of $2_{K2}$-free graphs"