Locally countable graphs of second projective class not generated by countably many projective functions

To answer a question by Rettich and Serafin, we define a model of set theory in which there exists a locally countable ${\Pi }_2^1$ graph on a subset of the real line, which is not generated by a countable family of projective (or even real-ordinal definable, ROD) functions. We also prove that the ${\Sigma }_2^1$ equi-constructibility graph on the reals is not generated by a countable family of ROD functions in the Solovay model.