Forbidden subgraphs for supereulerian and hamiltonian graphs

A graph is called supereulerian if it has a spanning eulerian subgraph. A graph is said to be hamiltonian if it has a spanning cycle. A nontrivial path is called a branch if it has only internal vertices of degree two and end vertices of degree not two. Let S be a set of branches of G, then S is called a branch cut if G−S has more components than G. A minimal branch cut is called a branch-bond. In this paper, we characterize one or pairs of those forbidden subgraphs that force a 2-edge-connected graph satisfying that every odd branch-bond has an edge branch to be supereulerian. We also characterize one or pairs of those forbidden subgraphs that force a 2-connected supereulerian graph to be hamiltonian.