def get_children_graph(self, item_ids=None, language=None, forbidden_item_ids=None):
        """
        Get a subgraph of items reachable from the given set of items through
        the 'child' relation.

        Args:
            item_ids (list): items which are taken as roots for the reachability
            language (str): if specified, filter out items which are not
                available in the given language

        Returns:
            dict: item id -> list of items (child items), root items are
            referenced by None key
        """
        if forbidden_item_ids is None:
            forbidden_item_ids = set()

        def _children(item_ids):
            if item_ids is None:
                items = Item.objects.filter(active=True).prefetch_related('children')
            else:
                item_ids = [ii for iis in item_ids.values() for ii in iis]
                items = Item.objects.filter(id__in=item_ids, active=True).prefetch_related('children')
            return {
                item.id: sorted([
                    _item.id for _item in item.children.all()
                    if _item.active and _item.id not in forbidden_item_ids
                ])
                for item in items if item.id not in forbidden_item_ids
            }

        if item_ids is None:
            return self._reachable_graph(None, _children, language=language)
        else:
            graph = self.get_children_graph(None, language, forbidden_item_ids=forbidden_item_ids)
            return self._subset_graph(graph, set(item_ids) - set(forbidden_item_ids))
Get a subgraph of items reachable from the given set of items through
        the 'child' relation.

        Args:
            item_ids (list): items which are taken as roots for the reachability
            language (str): if specified, filter out items which are not
                available in the given language

        Returns:
            dict: item id -> list of items (child items), root items are
            referenced by None key