Geodesic Convexity and Hop Domination in Graph Powers: The Path Family

Graph powers alter distances in a predictable manner, making them a useful setting for domination parameters defined through distance. We consider convex hop domination in powers of paths. The usual distance identity for graph powers is employed only as a metric reduction: distance two in the powered graph corresponds to the condition k dG(u,v) 2k in the original graph. Our contribution is the complete exact classification for path powers with k 2, supported by matching upper and lower bounds in the complete, diameter-two transition, and stable ranges. The case k = 1 is not new; it is the path result of Hassan, Canoy, and Saromines (2023) and is retained only to present the classification in a unified form. For higher powers, the argument relies on k-step geodesic progressions, endpoint-span estimates, and common-neighbor structure in the transition range. We also check the formulas exhaustively for 102 pairs with 1 k 6 and 2 n 18. The verification procedure, complete tested values, and executable Python code are supplied to make this computational check reproducible.