Hamilton cycles in 3-connected claw-free and net-free graphs

For an integer s1, s2, s3 > 0, let Ns1,s2,s3 denote the graph obtained by identifying each vertex of a K3 with an end vertex of three disjoint paths Ps1+1, Ps2+1, Ps3+1 of length s1, s2, and s3, respectively. We determine a family F of graphs such that, every 3-connected (K1,3, Ns1,s2,1)-free graph Γ with s1 + s2 + 1 ≤ 10 is hamiltonian if and only if the closure of Γ is L(G) for some graph G / ∈ F . We also obtain the following results.