Time exceeded error. What I can fix?

import heapq
import sys


class City:

    def __init__(self, city_id):
        self.city_id = city_id
        self.name = None
        self.key = float('inf')
        self.edge_list = list()
        self.parent = None
        self.visited = False

        
    def add_neighbor(self, edge):
        self.edge_list.append(edge)

    def __lt__(self, other):
        return self.key < other.key

    def is_not_visited(self):
        if self.visited is False:
            return True
        return False

class Edge:
    
    def __init__(self, to_city, cost):
        self.to_city = to_city
        self.cost = cost


class SaveList:

    def __init__(self, city_list, start, end):
        self.city_list = city_list
        self.start = start
        self.end = end


def daikstra(one_list, start, end):
    for one in one_list:
        one.visited = False
        one.key = float('inf')

    queue = []
    visited_list = list()
    start_city = None
    end_city = None
    for one in one_list:
        if one.name == start:
            start_city = one
        if one.name == end:
            end_city = one

    start_city.key = 0
    heapq.heappush(queue, start_city)
    while queue:
        current = heapq.heappop(queue)
        if current.visited: continue

        for edge in current.edge_list:
            if edge.to_city.is_not_visited:
                heapq.heappush(queue, edge.to_city)
                if current.key + edge.cost < edge.to_city.key:
                    edge.to_city.key = current.key + edge.cost
                    edge.to_city.parent = current
        current.visited = True
        visited_list.append(current)
        if current == end_city:
            break

    for one in visited_list:
        if one.city_id == end_city.city_id:
            return one.key




def main():
    case_list = list()
    case_number = int(sys.stdin.readline().rstrip())
    for c_num in range(0, case_number):
        city_list = list()
        city_number = int(sys.stdin.readline())
        for c in range(0, city_number):
            city = City(c)
            city_list.append(city)

        for city_num in range(0, city_number):
            city_name = str(sys.stdin.readline().rstrip('\n'))
            city_name = city_name.lower()
            neigh_number = int(sys.stdin.readline().rstrip('\n'))
            for nei_num in range(0, neigh_number):
                nei_cost = sys.stdin.readline()
                nei_cost = nei_cost.split(" ")
                neighbor = int(nei_cost[0])-1
                cost = int(nei_cost[1])
                city_list[city_num].name = city_name
                edge = Edge(city_list[neighbor], cost)
                city_list[city_num].add_neighbor(edge)
        num_paths = int(sys.stdin.readline())
        for n_path in range(0, num_paths):
            one_pair = sys.stdin.readline()
            one_pair = one_pair.split(" ")
            from_city = one_pair[0].rstrip()
            to_city = one_pair[1].rstrip()
            save_list = SaveList(city_list, from_city, to_city)
            case_list.append(save_list)

        skip = sys.stdin.readline()
    case_count = 0
    for one_list in case_list:
        value = daikstra(one_list.city_list, one_list.start, one_list.end)
        print(value)
        # sys.stdout.write("{0}".format(value))
        # if case_count < case_number:
        #     print()
        # case_count = case_count + 1


if __name__ == "__main__":
    main()