  We give a construction to remove coincidence points of continuous maps on
graphs (1-complexes) by changing the maps by homotopies. When the codomain is
not homeomorphic to the circle, we show that any pair of maps can be changed by
homotopies to be coincidence free. This means that there can be no nontrivial
coincidence index, Nielsen coincidence number, or coincidence Reidemeister
trace in this setting, and the results of our previous paper "A formula for the
coincidence Reidemeister trace of selfmaps on bouquets of circles" are invalid.
