Closure theorems in hamiltonian graph theory are of the following type: Let G be a 2- connected graph and let u, v be two distinct nonadjacent vertices of G. If condition c( u, v) holds, then G is hamiltonian if and only if G + uv is hamiltonian. We discuss several results of this type in which u and v are vertices of a subgraph H of G on four vertices and c( u, v) is a condition on the neighborhoods of the vertices of H (in G). We also discuss corresponding sufficient conditions for hamiltonicity of G.