Forbidden subgraphs that imply k-ordered and k-ordered hamiltonian

For a positive integer k, a graph G is k-ordered if for every ordered sequence of k vertices, there is a cycle that encounters the vertices in the sequence in the given order. If the cycle is also a hamiltonian cycle, then G is said to be k-ordered hamiltonian. Forbidden connected subgraphs and forbidden pairs of connected subgraphs that imply that a 2-connected graph is hamiltonian have been characterized. Each of these forbidden subgraph conditions will be investigated to determine if it implies more than just hamiltonicity, but in fact it implies k-ordered or k-ordered hamiltonian in the presence of the appropriate connectivity on the graph. More general classes of forbidden subgraphs that imply k-ordered and k-ordered hamiltonian will also be considered.