Find circle in a graph from a start point

cyclepath.py

def depth_first_search(map, key, start):
    if key == None: return []
    key = key.lower() if key.lower() in map else key.upper()
    if not key in map: return []
    if map[key] == None: return []
    res = []
    for point in map[key]:
        if point.lower() == start.lower():
            return [point]
        path = depth_first_search(map, point, start)
        if len(path) != 0:
            for circle in path:
                res.append(point + circle)
    return res


def find_cycle_in_path(map, start):
    if map == None: return []
    start = start.lower() if start.lower() in map else start.upper()
    if not start in map: return []
    if map[start] == None: return []
    res = []
    for point in map[start]:
        search = depth_first_search(map, point, start)
        if len(search) != 0:
            for circle in search:
                res.append(start + point + circle)
    return res