The problem of making bounded in-degree and out-degree data structures partially persistent is considered. The node copying method of Driscoll et al. is extended so that updates can be performed in worst-case constant time on the pointer machine model. Previously it was only known to be possible in amortised constant time.The result is presented in terms of a new strategy for Dietz and Raman's dynamic two player pebble game on graphs.It is shown how to implement the strategy and the upper bound on the required number of pebbles is improved from 2b+2d+O(√b) to d+2b. where b is the bound of the in-degree and d the bound of the out-degree. We also give a lower bound that shows that the number of pebbles depends on the out-degree d.