Reduced 2-to-1 maps and decompositions of graphs with no 2-to-1 cut sets

A graph has an increasing ear decomposition if it can be constructed from a simple closed curve by attaching arcs in stages with the endpoints of each arc attached to different points so that at least one new branch point is formed at each stage. A reduced 2-to-1 map is a 2-to-1 map that does not have a restriction that is 2-to-1. A 2-to-1 cut set of a graph G is a finite subset B such that G⧹B has at least 2|B| components. A graph has an increasing ear decomposition if and only if it does not have a 2-to-1 cut set, and a graph is the image of a reduced 2-to-1 map if and only if it does not have a 2-to-1 cut set.