In this paper we study the problem of deciding if, for a fixed graph H, a given graph is switching-equivalent to an H-free graph. Polynomial-time algorithms are known for H having at most three vertices or isomorphic to P ₄. We show that for H isomorphic to a claw, the problem is polynomial, too. Further, we give a characterization of graphs switching-equivalent to a K _1,2-free graph by ten forbidden induced subgraphs, each having five vertices. We also give the forbidden induced subgraphs for graphs switching-equivalent to a forest of bounded vertex degrees.