Spanning trees with at most 3 leaves in -free graphs

We present a solution of the following problem posed by Flandrin, Kaiser, Kuzel, Li and Ryjáček [E. Flandrin, T. Kaiser, R. Kuzel, H. Li and Z. Ryjáček, Discrete Math. 308(2008), 2343–2350]. very connected K 1 , 4 -free graph G with σ 4 ( G ) ≥ | G | contain a spanning tree with at most 3 leaves? σ 4 ( G ) = min { ∑ i = 1 4 deg G ( v i ) : { v 1 , v 2 , v 3 , v 4 }  is an independent set of  G } and K 1 , 4 -free graph is a graph without an induced K 1 , 4 subgraph.