Abstract. 

Hop domination was introduced as a distance-two analogue of domination and has been studied extensively in recent years. A secure hop dominating set, recently introduced, models a single adversarial attack at an unoccupied vertex (a vertex not in the current guard set) that can be defended by relocating one guard at distance two while preserving hop domination. Motivated by finite-order (multi-step) protection in classical secure domination, we introduce t-secure hop dominating sets (t ∈ N0), in which an adversary may launch a sequence of at most t attacks, each at a currently unoccupied vertex, and the defender responds by sequentially relocating one guard at distance two after each attack while maintaining hop domination throughout. Our main contribution is an exact correspondence: t-secure hop domination in a graph G is equivalent to smart t-secure domination in the hop graph H(G). This yields structural properties (monotonicity and additivity over components) and exact values for several graph families, including complete multipartite graphs, stars, paths, and cycles. In particular, we obtain closed formulas for γsh,t(Pn) and γsh,t(Cn) for all t ∈ N0, with explicit small-n exceptions in the cycle case.