Complexity of a reaching-a-node 2 player graph game

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The game:

  • There is an undirected simple graph with some special, winning nodes, and 2 players, who take turns.
  • Player one (Seeker) moves on the graph (sits on a node, and moves along an edge), and tries to reach a winning node. If she reaches one, she wins. Her initial position is given at the start.
  • Player two (Eraser) can turn the winning nodes into regular nodes. She wins, if the other player can't win, that is, she erased all the winning nodes into regular ones.

I am interested about this game.

E.g. if this game "can be solved" in polynomial or even in NP or coNP time, that is, whether it is possible to decide who has a winning strategy. Or whether this game is in the literature, and is known to be easy or hard, possibly in a different form.

Thank you!