In this paper, we consider a linear time-invariant discrete-time system and study the output null controllability problem, i.e., the problem of steering the output to zero in a finite number of steps. We assume that we only know the structure of the system, i.e., the zero/nonzero location in the system matrices. Hence, we consider a structural version of the output null controllability problem. We represent the structure of the system by means of a directed graph and present a graph theoretic sufficient condition for the problem to be generically solvable. Here generically solvable means that the problem is solvable for almost all systems with the same structure. We illustrate the conditions using an example.