Closure concepts for claw-free graphs

Recently, Ryjáček introduced an interesting new closure concept for claw-free graphs, and used it to prove that every nonhamiltonian claw-free graph is a spanning subgraph of a nonhamiltonian line graph (of a triangle-free graph). We discuss the relationship between Ryjáčekʹs closure and the K4-closure introduced by the first author. Our main result deals with a variation on the K4-closure. It implies a simpler proof of Ryjáčekʹs closure theorem, and yields a more general closure concept which is not restricted to claw-free graphs only.