Reduced k -to-1 maps and graphs with no k -to-1 cut set

A reduced map is a continuous function such that the preimage of every proper subcontinuum of the range is disconnected. A k -to-1 cut set is a finite subset B of a continuum X such that X\B has at least k|B| components. If a continuum is the image of a reduced at most k -to-1 map from a continuum, then it does not contain a k -to-1 cut set. A connected graph is the image of a reduced at most k -to-1 map from a continuum if and only if the graph does not contain a k -to-1 cut set. A connected graph is the image of a reduced k -to-1 map from a continuum if and only if the graph does not contain a k -to-1 cut set or an endpoint.