Nowadays, the operational management of complex water networks must be sustainable, safe and secure. It is therefore important to be able to constrain the state variables of the system as much as possible and necessary. The authors have recently developed a comprehensive framework that includes linkflow and outflow min-max constraints and pressure control, and which finds a steady-state solution without using any heuristics. A fixed-point nonlinear system is derived from Nash equilibrium theory, which include best-response functions and constraints. Some complementary developments are necessary to the solution method. On the way to the solution, there might be some disconnected nodes causing the iteration matrix to be rank-deficient; a regularization technique can be used to artificially reconnect the full system; there might be a disadvantage in that more iterations are needed due to the perturbation of the original solution method (e.g., the Newton method). In this communication, the authors will show how the regularization can be avoided in terms of graph analogy and equivalent equations. Highlights: - A general framework suitable for sustainable management of pressures and flow rates - No heuristics are required to determine the status of the pressure and Jow control devices - A reduced formulation is sought to benefit from Newton's quadratic convergence - The formulation takes into account min-max constraints for arc flows and outgoing node flows