#Partial






    # Generate all possible combinations of K linear orderings

    # Check if each ordering satisfies the transitive property



# Example usage

    # Example instance

    # Solve the "Partial Order Dimension" problem

    # Print the result


if __name__ == '__main__':
    Order
    dimension
    import itertools
    def is_transitive(DAG, ordering):
        for u, v in itertools.combinations(ordering, 2):
            if (u, v) not in DAG.edges():
                return False
        return True
    def partial_order_dimension(DAG, K):
        n = len(DAG.nodes())
        vertices = list(DAG.nodes())
        orderings = list(itertools.permutations(vertices, K))
        for ordering in orderings:
            if is_transitive(DAG, ordering):
                return True
        return False
    if __name__ == '__main__':
        import networkx as nx
        DAG = nx.DiGraph()
        DAG.add_edges_from([(1, 2), (2, 3), (1, 3)])
        K = 2
        result = partial_order_dimension(DAG, K)
        if result:
            print("DAG has a collection of linear orderings satisfying the constraints")
        else:
            print("DAG does not have a collection of linear orderings satisfying the constraints")
