Subgraph isomorphism in graph classes

We investigate the computational complexity of the following restricted variant of Subgraph Isomorphism: given a pair of connected   graphs G=(VG,EG)G=(VG,EG) and H=(VH,EH)H=(VH,EH), determine if HH is isomorphic to a spanning   subgraph of GG. The problem is NP-complete in general, and thus we consider cases where GG and HH belong to the same graph class such as the class of proper interval graphs, of trivially perfect graphs, and of bipartite permutation graphs. For these graph classes, several restricted versions of Subgraph Isomorphism such as Hamiltonian Path, Clique, Bandwidth, and Graph Isomorphism can be solved in polynomial time, while these problems are hard in general.