12-2.py

#!/usr/bin/env python

class Map:
    def __init__(self, input):
        self.neighbours = {}
        self.paths = -1

    def connect(self, cave1, cave2):
        n1 = self.neighbours.get(cave1, [])
        n2 = self.neighbours.get(cave2, [])

        if cave2 not in n1:
            n1.append(cave2)
        if cave1 not in n2:
            n2.append(cave1)

        self.neighbours[cave1] = n1
        self.neighbours[cave2] = n2

    def find_duplicate(self, l):
        for c in l:
            if c == c.casefold() and l.count(c) > 1:
                return True
        return False

    def visit(self, cave, path):
        if cave == 'end':
            self.paths += 1
            return

        next_path = path + [cave]
        for n in self.neighbours[cave]:
            if n == 'start':
                continue
            elif n == n.casefold() and \
                 next_path.count(n) >= 1 and \
                 self.find_duplicate(next_path):
                continue
            else:
                self.visit(n, next_path)

    def count_paths(self):
        if self.paths >= 0:
            return self.paths

        self.paths = 0
        self.visit('start', [])
        return self.paths

    def __str__(self):
        s = f"<{self.__class__.__name__} ({self.paths}):\n"
        for cave, neighbours in self.neighbours.items():
            s += f"{cave} -> {neighbours}\n"
        s += ">"
        return s


map = Map(input)

input = [list(line.split("-")) for line in open('12.input').readlines()]
for conn in input:
    map.connect(conn[0].strip(), conn[1].strip())

print(map.count_paths())