The s-Hamiltonian index

For integers k, s with 0 ≤ k ≤ s ≤ |V (G)| − 3, a graph G is called s-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 K1,3 and an integer m ≥ 0, we define hs(G) = min{m : L(G) is s-Hamiltonian} and l(G) = max{m : G has an arc of length m that is not both of length 2 and in a K3}, where an arc in G is a path in G whose internal vertices have degree two in G. We prove that hs(G) ≤ l + s+ 1.