Induced S(K1,3) and hamiltonian cycles in the square of a graph

We prove that the square of a connected graph in which every induced S(K1,3) has at least three edges in a block of degree at most 2 is hamiltonian. We also show that insertion of a block of degree 2 into, or under certain conditions also deletion of such a block, from a connected graph does not change the hamiltonicity of the square of the graph.