Problem: You are given an undirected unweighted graph with N vertices and M edges that contains neither self-loops nor double edges.   Here, a _self-loop_ is an edge where a_i = b_i (1≤i≤M), and _double edges_ are two edges where (a_i,b_i)=(a_j,b_j) or (a_i,b_i)=(b_j,a_j) (1≤i<j≤M).   How many different paths start from vertex 1 and visit all the vertices exactly once?   Here, the endpoints of a path are considered visited.  For example, let us assume that the following undirected graph shown in Figure 1 is given.    Figure 1: an example of an undirected graph  The following path shown in Figure 2 satisfies the condition.    Figure 2: an example of a path that satisfies the condition  However, the following path shown in Figure 3 does not satisfy the condition, because it does not visit all the vertices.    Figure 3: an example of a path that does not satisfy the condition  Neither the following path shown in Figure 4, because it does not start from vertex 1.    Figure 4: another example of a path that does not satisfy the condition

Submission attempts (in order):
def abc054_c_one_stroke_paths():
    import copy
    N, E = map(int, input().split())

    # create graph G
    G = {}
    for n in range(N):
        a, z = map(int, input().split())
        G[a] = G.get(a, [])
        G[a].append(z)
        G[z] = G.get(z, [])
        G[z].append(a)

    print(G)

    # find all paths
    if G.get(1) is None:
        print(0)
        return

    paths = []

    def solve(a, path):
        path.append(a)
        if len(path) == N:
            paths.append(path)
            return
        for z in G.get(a):
            p = copy.copy(path)
            print(p, a, z)
            if z in p:
                continue
            solve(z, p)

    solve(1, [])
    print(paths, len(paths))


##########
if __name__ == "__main__":
    abc054_c_one_stroke_paths()


def abc054_c_one_stroke_paths():
    import copy
    N, E = map(int, input().split())

    # create graph G
    G = {}
    for m in range(M):
        a, z = map(int, input().split())
        G[a] = G.get(a, [])
        G[a].append(z)
        G[z] = G.get(z, [])
        G[z].append(a)

    # print(G)

    # find all paths
    if G.get(1, 0) == 0:
        print(0)
        return

    paths = []

    def solve(a, path):
        path.append(a)
        if len(path) == N:
            paths.append(path)
            return
        for z in G.get(a):
            p = copy.copy(path)
            if z in p:
                continue
            solve(z, p)

    solve(1, [])
    # print(paths)
    print(len(paths))


##########
if __name__ == "__main__":
    abc054_c_one_stroke_paths()


def abc054_c_one_stroke_paths():
    N, M = map(int, input().split())

    # create graph G
    G = {}
    for m in range(M):
        a, z = map(int, input().split())
        G[a] = G.get(a, [])
        G[a].append(z)
        G[z] = G.get(z, [])
        G[z].append(a)

    # print(G)

    # find all paths
    if G.get(1, 0) == 0:
        print(0)
        return

    def dfs(a, path):
        path.append(a)
        if len(path) == N:
            return 1

        paths = 0
        for z in G.get(a):
            if z in path:
                continue
            paths += dfs(z, path[:])

        return paths

    print(dfs(1, []))


##########
if __name__ == "__main__":
    abc054_c_one_stroke_paths()


Verdicts: Runtime Error

Runtime Error

Accepted