#Rural Postman



    # Add required edges with high weight

    # Find the optimal solution using minimum-weight perfect matching



    # Find an Eulerian circuit in the matched graph


# Example usage:



if __name__ == '__main__':
    import networkx as nx
    def rural_postman(graph, required_edges):
        augmented_graph = graph.copy()
        for edge in required_edges:
            u, v, weight = edge
            augmented_graph.add_edge(u, v, weight=weight)
        matching_graph = nx.Graph()
        for u, v, weight in augmented_graph.edges(data='weight', default=1):
            matching_graph.add_edge(u, v, weight=weight)
        matching = nx.max_weight_matching(matching_graph, maxcardinality=True)
        matched_graph = augmented_graph.copy()
        for u, v in matching:
            matched_graph.add_edge(u, v, weight=matching_graph[u][v]['weight'])
        eulerian_circuit = nx.eulerian_circuit(matched_graph)
        return eulerian_circuit
    graph = nx.Graph()
    graph.add_edge('A', 'B', weight=2)
    graph.add_edge('B', 'C', weight=3)
    graph.add_edge('C', 'D', weight=4)
    graph.add_edge('D', 'A', weight=5)
    graph.add_edge('D', 'B', weight=1)
    required_edges = [('A', 'B', 2), ('D', 'B', 1)]
    path = rural_postman(graph, required_edges)
    total_cost = sum(graph[u][v]['weight'] for u, v in path)
    print("Rural Postman Path:", path)
    print("Total Cost:", total_cost)
