Reduced 2-to-1 maps and decompositions of graphs with no 2-to-1 cut sets (Q1862028)

\textit{J. Heath} [Continua with the Houston problem book, Lect. Notes Pure Appl. Math. 170, 89-102, Marcel Dekker, N.Y. (1995; Zbl 0852.54013)] has asked: given an integer \(k> 0\) and two graphs \(G\) and \(H\), when does there exist a \(k\)-to-\(1\) (continuous) map from \(G\) onto \(H\)? Here, a graph is a continuum which is a finite union of arcs with a finite number of points having order greater than two. The author has previously shown that a graph is the image of a tree under a \(2\)-to-\(1\) map with no \(2\)-to-\(1\) restriction to a continuum iff the graph satisfies a graph connectivity condition (``\(k\)-to-\(1\) cut set''). In this paper the author characterizes such graphs in terms of a graph decomposition property (``increasing ear decomposition'').