The ss-Hamiltonian index

For integers k,sk,s with 0⩽k⩽s⩽|V(G)|-30⩽k⩽s⩽|V(G)|-3, a graph GG is called ss-Hamiltonian if the removal of any k   vertices results in a Hamiltonian graph. For a simple connected graph that is not a path, a cycle or a K1,3K1,3 and an integer s⩾0s⩾0, we define hs(G)=min{m:Lm(G)iss-Hamiltonian} and l(G)=max{m:Gl(G)=max{m:G has a divalent path of length mm that is not both of length 2 and in a K3}K3}, where a divalent path in GG is a non-closed path in GG whose internal vertices have degree 2 in GG. We prove that hs(G)⩽l(G)+s+1hs(G)⩽l(G)+s+1.