Two edges of a graph are said to form a couple when their nodes can be labelled A, B, C, D so that A is adjacent to B but not to C, and D is adjacent to C but not to B. A graph is called matroidal if the binary relation “equals to or forms a couple with” among its edges is an equivalence relation. The structure of matroidal graphs is determined. The smallest number of inequalities necessary to separate the incidence vectors of the node packings of a matroidal graph from the other incidence vectors is equal to the size of the largest equivalence class of the above relation. This number is not larger than one half of the number of nodes. All matroidal graphs are perfect.