Determine All Players with Fixed Rankings

Problem Statement

There are n players labeled from 1 to n participating in a round-robin style competition. You are given an integer n and a 2D array matches where matches[i] = [u, v] indicates that player u defeated player v (u is ranked strictly higher than v).

The ranking relation is strictly transitive: if player A defeats player B and player B defeats player C, then player A is ranked higher than player C.

A player's exact rank can be uniquely determined if and only if their relative standing against every other player is known (i.e., for every other player x, we can deduce either that p is ranked higher than x or that x is ranked higher than p).

Assume the match results are consistent and contain no contradictions (cycles). Return a sorted list of all player IDs whose exact final ranking can be uniquely determined.

Function Signature

def find_fixed_rank_players(n: int, matches: list[list[int]]) -> list[int]:
    pass

Constraints

• 1 <= n <= 500
• 0 <= matches.length <= 10^4
• matches[i] = [u, v] with 1 <= u, v <= n and u != v
• All pairs in matches are distinct, and there are no cyclic contradictions.

Examples

Example 1

Input: n = 5, matches = [[4, 3], [4, 2], [3, 2], [1, 2], [2, 5]]
Output: [2]
Explanation:
- Player 2 is beaten by 4, 3, and 1, and beats 5.
  Player 2 has relations with all 4 other players (3 above, 1 below), so player 2's rank is uniquely 4th.
- For player 1, we know 1 > 2 and 1 > 5, but relative ordering between 1, 3, and 4 is unknown.
Only player 2 has a fixed rank.

Example 2

Input: n = 3, matches = [[1, 2], [2, 3]]
Output: [1, 2, 3]
Explanation:
1 > 2 > 3 forms a total order. Every player's rank is uniquely determined:
Player 1 is 1st, player 2 is 2nd, player 3 is 3rd.

Edge Cases

• n = 1: The single player always has rank 1 ([1]).
• matches is empty and n > 1: No relationships exist; return [].
• Disconnected components: Players with no connections cannot have fixed ranks unless n = 1.