Problem:
We have a permutation of the integers from 1 through N, p_1, p_2, .., p_N. We also have M pairs of two integers between 1 and N (inclusive), represented as (x_1,y_1), (x_2,y_2), .., (x_M,y_M). AtCoDeer the deer is going to perform the following operation on p as many times as desired so that the number of i (1 ≤ i ≤ N) such that p_i = i is maximized:    * Choose j such that 1 ≤ j ≤ M, and swap p_{x_j} and p_{y_j}.  Find the maximum possible number of i such that p_i = i after operations.

Incorrect submission (verdict: Wrong Answer):
import sys

sys.setrecursionlimit(2000000000)


class Node:
    def __init__(self, index):
        self.index = index
        self.neighbors = set()

    def __repr__(self):
        return '{}: {}'.format(self.index, self.neighbors)


def can_go_to_dst(nodes, src_index, dst_index, past_indices):
    if nodes[src_index] is None:
        return False

    neighbor_indices = nodes[src_index].neighbors - past_indices
    if dst_index in neighbor_indices:
        return True

    return any(
        [can_go_to_dst(nodes, ni, dst_index, past_indices | {src_index})] for ni
        in neighbor_indices)


# line_index = 0
#

#     global line_index

#              '5 3 6 8 7 10 9 1 2 4',
#              '3 1',
#              '4 1',
#              '5 9',
#              '2 5',
#              '6 5',
#              '3 5',
#              '8 9',
#              '7 9']
#
#     line = lines[line_index]
#     line_index += 1



if __name__ == '__main__':
    N, M = map(int, input().split())

    p = []
    desirable_swaps = []
    for i, s in enumerate(input().split()):
        s_value = int(s) - 1
        if i != s_value:
            desirable_swaps.append((s_value, i))

    nodes = [None] * N
    for _ in range(M):
        xy_str = input().split()
        x, y = int(xy_str[0]) - 1, int(xy_str[1]) - 1
        if nodes[x] is None:
            nodes[x] = Node(x)
        if nodes[y] is None:
            nodes[y] = Node(y)

        nodes[x].neighbors |= {y}
        nodes[y].neighbors |= {x}

    first_n_equals = N - len(desirable_swaps)

    n_op_equals = 0
    for src_index, dst_index in desirable_swaps:
        if can_go_to_dst(nodes, src_index, dst_index, set()):
            n_op_equals += 1

    answer = first_n_equals + n_op_equals
    if answer % 2 == 1:
        answer -= 1
    print(answer)
    
    # print(nodes)


Fixed, accepted submission:
import sys

sys.setrecursionlimit(2000000000)


class Node:
    def __init__(self, index):
        self.index = index
        self.connected_indices = set()


def build_graph(nodes, curr_index, connected_indices, reached_indices):
    # already reached
    if curr_index in connected_indices or curr_index in reached_indices:
        return

    connected_indices |= {curr_index}
    reached_indices |= {curr_index}

    unreached_indices = nodes[curr_index].connected_indices - connected_indices

    for i in unreached_indices:
        build_graph(nodes, i, connected_indices, reached_indices)

    return connected_indices


if __name__ == '__main__':
    N, M = map(int, input().split())
    p = [int(v) - 1 for v in input().split()]

    nodes = [Node(i) for i in range(N)]
    for _ in range(M):
        xy_str = input().split()
        x, y = int(xy_str[0]) - 1, int(xy_str[1]) - 1

        # link neighbors
        nodes[x].connected_indices |= {y}
        nodes[y].connected_indices |= {x}

    reached_indices = set()
    all_connected_indices = [None] * N
    for i in range(N):
        connected_indices = build_graph(nodes, i, set(), reached_indices)
        if connected_indices is None:
            continue

        for c in connected_indices:
            all_connected_indices[c] = connected_indices

    answer = sum([p[i] in all_connected_indices[i] for i in range(N)])
    print(answer)