class GraphIsomorphism:
    """
    A simplified implementation of the VF2 algorithm for checking graph isomorphism.
    Graph isomorphism is the problem of determining whether two graphs are isomorphic,
    i.e., whether there is a one-to-one correspondence between their vertices that
    preserves adjacency.
    """
    def __init__(self, graph1, graph2):
        """
        Initializes the GraphIsomorphism object.

        Args:
            graph1 (dict): Adjacency list of the first graph.
            graph2 (dict): Adjacency list of the second graph.
        """
        self.g1 = graph1
        self.g2 = graph2
        self.v1 = list(graph1.keys())
        self.v2 = list(graph2.keys())

    def are_isomorphic(self):
        """
        Checks if the two graphs are isomorphic.

        Returns:
            bool: True if the graphs are isomorphic, False otherwise.
        """
        if len(self.v1) != len(self.v2):
            return False

        # A simple check: degree sequences must match
        d1 = sorted([len(self.g1[v]) for v in self.v1])
        d2 = sorted([len(self.g2[v]) for v in self.v2])
        if d1 != d2:
            return False

        # The core of VF2 is a recursive search with feasibility rules.
        # This is a simplified version for demonstration.
        # A full implementation is complex.
        mapping = {}
        return self._vf2_match(mapping)

    def _vf2_match(self, mapping):
        """
        Recursive matching function for VF2.
        """
        if len(mapping) == len(self.v1):
            return True

        # Get the next candidate pair to add to the mapping
        u, v = self._next_candidate_pair(mapping)

        if u is None or v is None:
            return False

        for v_candidate in self._get_feasible_v_candidates(u, mapping):
            if self._is_feasible(u, v_candidate, mapping):
                new_mapping = mapping.copy()
                new_mapping[u] = v_candidate
                if self._vf2_match(new_mapping):
                    return True
        return False

    def _next_candidate_pair(self, mapping):
        # In a real VF2, this would be more intelligent.
        # Here, we just pick the next unmapped vertex from g1.
        u = next((vertex for vertex in self.v1 if vertex not in mapping), None)
        return u, None # v is determined by feasibility checks

    def _get_feasible_v_candidates(self, u, mapping):
        # Return all unmapped vertices in g2
        return [v for v in self.v2 if v not in mapping.values()]

    def _is_feasible(self, u, v, mapping):
        """
        Check if mapping u to v is feasible.
        """
        # Check if degrees are compatible
        if len(self.g1.get(u, [])) != len(self.g2.get(v, [])):
            return False

        # Check consistency with the current partial mapping
        for u_mapped, v_mapped in mapping.items():
            # Check if adjacency is preserved
            u_adj_to_u_mapped = u_mapped in self.g1.get(u, [])
            v_adj_to_v_mapped = v_mapped in self.g2.get(v, [])
            if u_adj_to_u_mapped != v_adj_to_v_mapped:
                return False
        return True
