  Parity games can be used to represent many different kinds of decision
problems. In practice, tools that use parity games often rely on a
specification in a higher-order logic from which the actual game can be
obtained by means of an exploration. For many of these decision problems we are
only interested in the solution for a designated vertex in the game. We
formalise how to use on-the-fly solving techniques during the exploration
process, and show that this can help to decide the winner of such a designated
vertex in an incomplete game. Furthermore, we define partial solving techniques
for incomplete parity games and show how these can be made resilient to work
directly on the incomplete game, rather than on a set of safe vertices. We
implement our techniques for symbolic parity games and study their
effectiveness in practice, showing that speed-ups of several orders of
magnitude are feasible and overhead (if unavoidable) is typically low.
