A \(\Lambda\) in a graph \(X\) is two edges \(uv\) and \(vw\) such that \(uw\) is not an edge. A subgraph \(Y\) of \(X\) is called a \(\Lambda\)-subgraph if every \(\Lambda\) of \(X\) has both or neither of its edges in \(Y\). Let \(X\) and \(Y\) be two vertex-disjoint graphs, and \(v\) be a vertex of \(X\). The substitution of graph \(Y\) for vertex \(v\) in \(X\) is obtained by deleting the vertex \(v\) (and all incident edges) from \(X\) and making every vertex of \(Y\) adjacent to each neighbor of \(v\) in \(X\). Duplication of the vertex \(v\in X\) \(c\) times is obtained by letting \(Y=\overline{K_c}\). The authors prove three main results and use them to prove a decomposition theorem of T. Gallai. The first main result states that if \({\mathcal C}\) is a non-empty class of graphs closed under substitution and taking induced subgraphs, and \(X\) is a union of \(\Lambda\)-subgraphs all of which are in \({\mathcal C}\), then \(X\) itself is in \({\mathcal C}\). The second result states that if \(X\) is in a class of graphs closed under duplicating vertices and taking induced subgraphs, then each prime \(\Lambda\)-subgraph of \(X\) is in \({\mathcal C}\). The third result states that if \({\mathcal C}\) is a non-empty class of graphs closed under substitution, complementation, and taking induced subgraphs, and if the edges of a complete graph are colored with three colors so that no triangle is orthogonal to the edge coloring, then if any two of the edge-induced monochromatic subgraphs are members of \({\mathcal C}\), then so is the third.