We discuss, on general grounds, how two subgraphs of a given Feynman graph can overlap with each other. For this, we use the notion of connecting and returning lines that describe how any subgraph is inserted within the original graph. This, in turn, allows us to derive "non-overlap" theorems for one-particle-irreducible subgraphs with $2$, $3$ and $4$ external legs. As an application, we provide a simple justification of the skeleton expansion for vertex functions with more than five legs, in the case of scalar field theories. We also discuss how the skeleton expansion can be extended to other classes of graphs.