In the book [Ll. Alseda, J. Llibre and M. Misiurewicz, Combinatorial Dynamics and Entropy in Dimension One] there is a proof that every continuous piecewise monotone interval map with positive entropy is semiconjugate to a map with constant slope. This proof is completely different from the classical proof of Milnor and Thurston. It turns out that a small modification of this proof works for piecewise continuous piecewise monotone interval maps. As a corollary we get a version of this theorem for (piecewise) continuous piecewise monotone graph maps.

The semiconjugacies that we consider are nondecreasing surjections. They may collapse some intervals to points. For graphs, this means that some nontrivial portions of the graph can be collapsed to the points, so the image of the graph can be a different graph.

This problem does not exist if the map is transitive. Then the semiconjugacy is a conjugacy. In particular, the graph we get after the semiconjugacy is homeomorphic to the original one.