A wounded graph W is a graph obtained from a K3 by adding a pendant edge at one vertex of the K3 and adding a path of length 2 at another vertex of the K3. A net N is a graph obtained from a K3 by adding a pendant edge at each vertex of the K3. A P6 is a path on 6 vertices. Čada et al. (2016) conjectured that for a 2-connected claw-free graph H and for a fixed graph S∈{W,N,P6}, if the degree at each end-vertex of every induced copy of S is at least (|V(H)|−2)/3, then H is Hamiltonian. The case for S=N was conjectured by Broersma in [2] (1993). The conjecture for S∈{N,P6} was proved in [9]. In this paper, we prove the conjecture for the last case S=W. A new and shorter proof for the case S=N is also included.