Minimum embedding of Steiner triple systems into (K4−e)(K4−e)-designs II

A (K4−e)(K4−e)-design of order v+wv+wembeds   a given Steiner triple system if there is a subset of vv points on which the graphs of the design induce the blocks of the original Steiner triple system. It has been established that w≥v/3w≥v/3, and that when equality is met, such a minimum embedding of an STS(vv) exists, except when v=15v=15. Equality only holds when v≡15,27(mod30). One natural question is: What is the smallest order ww such that some STS(v)(v) can be embedded into a (K4−e)(K4−e)-design of order v+wv+w? We solve the problem for 7 of the 10 congruence classes modulo 30.