#Rural Postman

import networkx as nx

def rural_postman(graph, required_edges):
    augmented_graph = graph.copy()

    # Add required edges with high weight
    for edge in required_edges:
        u, v, weight = edge
        augmented_graph.add_edge(u, v, weight=weight)

    # Find the optimal solution using minimum-weight perfect matching
    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'])

    # Find an Eulerian circuit in the matched graph
    eulerian_circuit = nx.eulerian_circuit(matched_graph)

    return eulerian_circuit

# Example usage:
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)
