Star connections

def other_points(point=None):
    for i in [i for i in range(9) if i != point]:
        yield i

def get_connections(points):
    connects = set()
    for i in range(len(points) - 1):
        connects |= connections[f({points[i], points[i + 1]})]
    return connects

f = frozenset

connections = {f({0, 2}): f({1}), f({0, 6}): f({3}), f({0, 8}): f({4}),
               f({1, 7}): f({4}), f({2, 6}): f({4}), f({2, 8}): f({5}),
               f({3, 5}): f({4})}

for start in range(9):
    for end in range(start, 9):
        fs = f({start, end})
        connections[fs] = connections.get(fs, f()) | fs

for start in range(9):
    for mid_1 in other_points(start):
        for mid_2 in other_points(mid_1):
            for mid_3 in other_points(mid_2):
                for end in other_points(mid_3):
                    points = (start, mid_1, mid_2, mid_3, end)
                    if get_connections(points) == set(range(9)):
                        print(points)