Claw-Free and $$N(2,1,0)$$ N ( 2 , 1 , 0 ) -Free Graphs are Almost Net-Free

In this paper, we characterize connected $$\{K_{1,3},N(2,1,0)\}$${K1,3,N(2,1,0)}-free but not $$N(1,1,1)$$N(1,1,1)-free graphs. By combining our result and a theorem showed by Duffus et al. (every $$2$$2-connected $$\{K_{1,3},N(1,1,1)\}$${K1,3,N(1,1,1)}-free graph is Hamiltonian), we give an alternative proof of Bedrossian's theorem (every $$2$$2-connected $$\{K_{1,3},N(2,1,0)\}$${K1,3,N(2,1,0)}-free graph is Hamiltonian).