In this paper, we study convergence theorems for finding a fixed point of a mapping with a directed graph. We give a counterexample to the result recently proved by Tripak (Fixed Point Theory Appl. 2016:87, 2016) and propose a better version of the theorem. In the presence of the transitivity of a directed graph, we prove convergence theorems without the uniform convexity of the space. Moreover, we obtain a sufficient condition for the existence of a fixed point of G-nonexpansive mappings. We also discuss a convergence theorem without assuming the transitivity on the graph G. In this setting, we obtain a result for a wider class of mappings including all G-nonexpansive mappings with a fixed point. Our results not only correct the original theorem, but also improve it by removing some of its assumptions.
