Cat and Mouse

Two players alternate moves along edges of an undirected graph: the mouse starts at 1, the cat at 2 (cat cannot enter node 0), and the game ends if the mouse reaches hole 0 (mouse wins), the cat lands on the mouse (cat wins), or a position repeats (draw). Determine which of the three outcomes occurs under optimal play — essentially resolving a finite game-state graph (mouse, cat, turn) with cycles/draws via retrograde analysis/BFS.