Before revision:  We show our result by proving  that for every k\geq 1, s\geq 1, t\geq 1, the class of (K_k,K_{1,s}^1,P_t)-free graphs has bounded mim-width and that a corresponding branch decomposition is "quickly computable"  .
After revision:  We show our result by proving boundedness of an underlying width parameter. Namely, we show that for every k\geq 1, s\geq 1, t\geq 1, the class of (K_k,K_{1,s}^1,P_t)-free graphs has bounded mim-width and that a corresponding branch decomposition is "quickly computable" .