Singular Turan numbers of stars

Suppose that G is a graph and H is a subgraph of G. We call H singular if the vertices of H either have the same degree in G or have pairwise distinct degrees in G. Let T-S(n, H) be the largest number of edges of a graph with n vertices that does not contain a singular copy of H. The problem of determining T-S(n, H) was studied initially by Caro and Tuza, who obtained an asymptotic bound for each H. In this paper, we consider the case that H is a star, and determine the exact values of T-S(n, K1,2) for all n, T-S(n, K1,4) and T-S(n, K-1,K-2s+1) for sufficiently large n.