The focus of sustainable urban transportation development lies in realizing the untapped capacity potential of the existing road network and enhancing its operational efficiency without expanding its physical footprint. To quantify the supply capacity of road networks in mountainous cities, this paper converts the problem of solving the capacity of road networks into the problem of solving the minimum cut set in road networks from the perspective of road network capacity, using the idea of the auxiliary diagram method in graph theory. By improving the limitation that the auxiliary map method is only applicable to the single starting point and terminal point network, a regional road network capacity estimation model of a mountain city based on the auxiliary map is constructed. Combined with the actual regional road network, the model results presented in this paper show that the road network capacity calculated by the auxiliary graph method is 30,137 pcu/h. Using the improved traffic distribution simulation method, the network capacity is 38,776 pcu/h. Compared with the traffic distribution simulation method, the regional road network capacity model based on an auxiliary map proposed in this paper is more realistic.