A game on an undirected graph is played by two players, Mouse and Cat, who alternate turns.
The graph is given as follows: graph[a] is a list of all nodes b such that ab is an edge of the graph.
Mouse starts at node 1 and goes first, Cat starts at node 2 and goes second, and there is a Hole at node 0.
During each player's turn, they must travel along one edge of the graph that meets where they are.  For example, if the Mouse is at node 1, it must travel to any node in graph[1].
Additionally, it is not allowed for the Cat to travel to the Hole (node 0.)
Then, the game can end in 3 ways:

If ever the Cat occupies the same node as the Mouse, the Cat wins.
If ever the Mouse reaches the Hole, the Mouse wins.
If ever a position is repeated (ie. the players are in the same position as a previous turn, and it is the same player's turn to move), the game is a draw.

Given a graph, and assuming both players play optimally, return 1 if the game is won by Mouse, 2 if the game is won by Cat, and 0 if the game is a draw.
 



Example 1:
Input: [[2,5],[3],[0,4,5],[1,4,5],[2,3],[0,2,3]]
Output: 0
Explanation:
4---3---1
|   |
2---5
 \ /
  0

 
Note:

3 <= graph.length <= 50
It is guaranteed that graph[1] is non-empty.
It is guaranteed that graph[2] contains a non-zero element.

```python

USER_FILENAME = "<user_code>"
import sys, json, builtins
import math, heapq, bisect, itertools, functools, collections, statistics, fractions, decimal, operator, array, random, re, string, os, pathlib
try:
    import resource as _res
except Exception:
    _res = None
from collections import Counter, defaultdict, deque, OrderedDict
from functools import lru_cache, reduce
from itertools import accumulate, permutations, combinations, product
from math import gcd, ceil, floor, sqrt
from decimal import Decimal
try:
    from typing import *
except Exception:
    pass
try:
    sys.setrecursionlimit(1_000_000)
except Exception:
    pass
try:
    import site as _site
    try:
        _sys_sites = _site.getsitepackages() if hasattr(_site, "getsitepackages") else []
        for _p in _sys_sites:
            if _p and _p not in sys.path:
                sys.path.append(_p)
    except Exception:
        pass
    try:
        _usp = _site.getusersitepackages()
        if isinstance(_usp, str):
            _usp = [_usp]
        for _p in _usp:
            if _p and _p not in sys.path:
                sys.path.append(_p)
    except Exception:
        pass
except Exception:
    pass

def _disabled_input(*args, **kwargs):
    raise RuntimeError("input() is disabled in callable mode")
builtins.input = _disabled_input

ns = {"__name__": "__judge__"}  # prevent if __name__ == "__main__" blocks
for _k, _v in list(globals().items()):
    if not _k.startswith("_"):
        ns[_k] = _v
_CODE = "class Solution:\n    # minimax\n    # https://www.youtube.com/watch?v=WJ7uqznd_4s\n    def catMouseGame(self, graph: List[List[int]]) -> int:\n        memo = {}\n\n        def mouse_play(key, cat, mouse, turn):\n            if any(v == 0 for v in graph[mouse]):\n                memo[key] = 1\n                return 1\n            res = 2\n            for nxt in graph[mouse]:\n                # mouse reach 0, mouse wins\n                # mouse would not run int place cat holds now\n                if nxt == cat:\n                    continue\n                tmp = move(cat, nxt, False)\n                # if mouse win, directly break, mouse would always take the optimal solution\n                if tmp == 1:\n                    res = 1\n                    break\n                if tmp == 0:\n                    res = 0\n            memo[key] = res\n\n            return res\n\n        def cat_play(key, cat, mouse, turn):\n            if any(v == mouse for v in graph[cat]):\n                memo[key] = 2\n                return 2\n            res = 1\n            for nxt in graph[cat]:\n                # if nxt == mouse:\n                #     memo[key] = 2\n                #     return 2\n                # cat can not enter hole\n                if nxt == 0:\n                    continue\n                tmp = move(nxt, mouse, True)\n                if tmp == 2:\n                    res = 2\n                    break\n                if tmp == 0:\n                    res = 0\n            memo[key] = res\n\n            return res\n\n        def move(cat, mouse, turn):\n            key = (cat, mouse, turn)\n            if key in memo:\n                return memo[key]\n\n            memo[key] = 0\n\n            if turn:\n                return mouse_play(key, cat, mouse, turn)\n            else:\n                return cat_play(key, cat, mouse, turn)\n\n        return move(2, 1, True)\n\n    def catMouseGame(self, graph: List[List[int]]) -> int:\n        N = len(graph)\n        DRAW, MOUSE, CAT = 0, 1, 2\n\n        def parents(m, c, t):\n            if t == MOUSE:\n                for c2 in graph[c]:\n                    if c2:  # cat can not enter 0\n                        yield m, c2, 3 - t\n            if t == CAT:\n                for m2 in graph[m]:\n                    yield m2, c, 3 - t\n\n        color, degree = collections.defaultdict(int), {}\n\n        # (mouse, cat, turn), got the parents/children of this node, [[2,3],...], since the graph is un-directed\n        # you can say 2,3 -> 0 (parents) or 0 -> (2,3) children, here we regard it as parents\n        for m in range(N):\n            for c in range(N):\n                degree[m, c, MOUSE] = len(graph[m])\n                degree[m, c, CAT] = len(graph[c]) - (0 in graph[c])\n\n        que = collections.deque()\n        for i in range(N):\n            for t in range(1, 3):\n                color[0, i, t] = MOUSE\n                que.append((0, i, t))\n                if i > 0:\n                    color[i, i, t] = CAT\n                    que.append((i, i, t))\n\n        while que:\n            m, c, t = que.pop()\n            col = color[m, c, t]\n            # for (m,c,t)'s parents, if\n            for m2, c2, t2 in parents(m, c, t):\n                if color[m2, c2, t2] is DRAW:\n                    if t2 == col:  # mouse 's turn and it goes to MOUSE, or cat's turn, go to CAT\n                        color[m2, c2, t2] = col\n                        que.append((m2, c2, t2))\n                    else:\n                        # (m2,c2,t2) is (m,c,t)'s parent, if cat went to MOUSE, or mouse went to CAT,\n                        # this is a lose situation, degree -=1, which means we have a losing for its\n                        # one children, if all chidren are losing for this parent, ie, degree == 0\n                        # then we add it to queue\n                        degree[m2, c2, t2] -= 1\n                        if not degree[m2, c2, t2]:\n                            color[m2, c2, t2] = 3 - t2\n                            que.append((m2, c2, t2))\n\n        return color[1, 2, 1]\n\n\n"
_co = compile(_CODE, USER_FILENAME, "exec")
exec(_co, ns, ns)

_target = None
_fn_name = 'catMouseGame'
if _fn_name:
    if "Solution" in ns:
        try:
            _obj = ns["Solution"]()
            if hasattr(_obj, _fn_name):
                _target = getattr(_obj, _fn_name)
        except Exception:
            _target = None
    if _target is None and _fn_name in ns and callable(ns[_fn_name]):
        _target = ns[_fn_name]
# Fallback: if there's exactly one user-defined function, use it
if _target is None:
    _cands = [v for k, v in ns.items() if callable(v) and getattr(v, "__module__", "") in ("__judge__", "__main__")]
    _user_funcs = [c for c in _cands if getattr(c, "__name__", "").startswith("_") is False]
    if len(_user_funcs) == 1:
        _target = _user_funcs[0]
if _target is None:
    raise RuntimeError(f"Could not resolve callable '{_fn_name}'")

_raw = sys.stdin.read().strip()
if _raw:
    _parsed = json.loads(_raw)
    if isinstance(_parsed, dict):
        _kwargs = _parsed.get("kwargs", {})
        _args = _parsed.get("args", [])
    else:
        _args, _kwargs = _parsed, {}
else:
    _args, _kwargs = [], {}
if not isinstance(_args, (list, tuple)):
    _args = [_args]
_res = _target(*_args, **_kwargs)
try:
    sys.stdout.write(repr(_res) + "\n")
except Exception:
    pass

```

Measured memory_bytes: 28144