Forbidden Subgraphs for Collapsible Graphs and Supereulerian Graphs

In this paper, we completely characterize the connected forbidden subgraphs and pairs of connected forbidden subgraphs that force a 2-edge-connected (2-connected) graph to be collapsible. In addition, the characterization of pairs of connected forbidden subgraphs that imply a 2-edge-connected graph of minimum degree at least three is supereulerian will be considered. We have given all possible forbidden pairs. In particular, we prove that every 2-edge-connected noncollapsible (or nonsupereulerian) graph of minimum degree at least three is Z₃-free if and only if it is K₃-free, where Z_i is a graph obtained by identifying a vertex of a K₃ with an end-vertex of a P_i+1.