Cat and Mouse

Alice controls the mouse and Bob controls the cat on an undirected graph with n nodes numbered from 0 to n-1. Node 0 is the mouse's safe house. Initially, the mouse is at node 1 and the cat is at node 2. Alice moves first. On each turn, a player may move their piece along an edge to an adjacent node or stay in place. The cat wins if it occupies the same node as the mouse (catching it). The mouse wins if it reaches node 0. If the cat can avoid losing indefinitely, the result is a draw (return false). Both play optimally — determine if Alice (mouse) can reach the safe house.

Examples
Input: [[1],[0,2],[1]]
Output: false
Hints

Cat and Mouse

Alice controls the mouse and Bob controls the cat on an undirected graph with n nodes numbered from 0 to n-1. Node 0 is the mouse's safe house. Initially, the mouse is at node 1 and the cat is at node 2. Alice moves first. On each turn, a player may move their piece along an edge to an adjacent node or stay in place. The cat wins if it occupies the same node as the mouse (catching it). The mouse wins if it reaches node 0. If the cat can avoid losing indefinitely, the result is a draw (return false). Both play optimally — determine if Alice (mouse) can reach the safe house.