class Node:
    def __init__(self, id, color):
        self.id = id # int
        self.color = color # string
        self.adjacentNodes = set()

# actually this function return the unique string associated with the graph
# that will be hashed with the function hash() in a second moment
def new_hash(graph, queue=[]): # graph: list of Node
    if not queue: # first call: find the root of the tree
        graph.sort(key = lambda x: (len(x.adjacentNodes), x.color), reverse=True)
        groups = itertools.groupby(graph, key = lambda x: (len(x.adjacentNodes), x.color))
        roots = []
        result_hash = ''

        for _, group in groups:
            roots  = [x for x in group]
            break # I just need the first (the candidates roots)

        temp_hashes = []

        for node in roots:
            temp_queue = [node.id]
            temp_hash = node.color + str(len(node.adjacentNodes)) + str(temp_queue.index(node.id))
            temp_hash += new_hash(list(node.adjacentNodes), temp_queue)
            temp_hashes.append((node, temp_hash, temp_queue))

        temp_hashes.sort(key = lambda x: x[1], reverse=True)
        queue = temp_hashes[0][2]
        result_hash += temp_hashes[0][1]
        result_hash += new_hash(list(temp_hashes[0][0].adjacentNodes), queue=queue)
    else:
        graph.sort(key = lambda x: (len(x.adjacentNodes), x.color), reverse=True)
        groups = itertools.groupby(graph, key = lambda x: (len(x.adjacentNodes), x.color))
        grouped_nodes = []
        result_hash = ''

        for _, group in groups:
            grouped_nodes.append([x for x in group])

        for group in grouped_nodes:
            while len(group) > 0:
                temp_hashes = []

                for node in group:
                    if node.id in queue:
                        temp_hash = node.color + str(len(node.adjacentNodes)) + str(queue.index(node.id))
                        temp_hashes.append((node, temp_hash, queue))
                    else:
                        temp_queue = queue[:]
                        temp_queue.append(node.id)
                        temp_hash = node.color + str(len(node.adjacentNodes)) + str(temp_queue.index(node.id))
                        temp_hash += new_hash(list(node.adjacentNodes), queue=temp_queue)
                        temp_hashes.append((node, temp_hash, temp_queue))

                temp_hashes.sort(key = lambda x: x[1], reverse=True)
                queue = temp_hashes[0][2]
                result_hash += temp_hashes[0][1]
                group.remove(temp_hashes[0][0])

    return result_hash