Forbidden set of induced subgraphs for 2-connected supereulerian graphs

Let H={H₁,…,H_k} be a set of connected graphs. A graph is said to be H-free if it does not contain any member of H as an induced subgraph. We show that if the following statements hold, • |H|≥2,• K_1,3∈H,• for any integer n₀, every 2-connected H-free graph G of order at least n₀ is supereulerian, i.e.,G has a spanning closed trailthen H∖{K_1,3} contains an N_i,j,k or a path where N_i,j,k denotes the graph obtained by attaching three vertex-disjoint paths of lengths i,j,k≥0 to a triangle. As an application, we characterize all the forbidden triples H with K_1,3∈H such that every 2-connected H-free graph is supereulerian. As a byproduct, we also characterize minimal 2-connected non-supereulerian claw-free graphs.