Graph theory provides a useful framework for generating insights into problems of sufficiency and optimality across a wide variety of physical relationships. Applied to the realm of grand strategy, this approach assists in developing a methodology for estimating the minimum level of forces required and determining the optimal deployments for the successful pursuit of national security goals. In theory, the adoption of a defense-in-depth maneuver strategy provides the most efficient use of scarce resources. However, deterrence stability attenuates due to the absence of robust local balances of forces. Comparative case analyses of the Roman and British empires confirm the efficiency of depth defense, as well as the weakening of deterrence. Implications for U.S. policy are that, despite sizeable reductions, two regional wars can be fought and won, nearly simultaneously, even below base force levels. However, the deployments required to effect this grand strategy may make challenges to conventional deterrence more likely. Finally, it is demonstrated that small increases in forces above minimum requirements create a valuable "margin of safety" and may significantly improve crisis and deterrence stability.