Detecting induced subgraphs

An s-graph is a graph with two kind of edges: subdivisible edges and real edges. A realisation of an s-graph B is any graph obtained by subdividing subdivisible edges of B into paths of length at least one. Given an s-graph B, we study the decision problem Π B . Its instance is any graph G, its question is “Does G contains a realisation of B as an induced subgraph ?”. For several Bʹs, the complexity is known and here we give the complexity for several more. We also provide results on the problem of detecting an induced cycle through two prescribed vertices. hs yielding trivially polynomial problems ds, prisms and thetas