Abstract:

Locating-hop domination refines hop domination by requiring vertices outside a hop dominating set to have nonempty and pairwise distinct distance-two signatures. Domination sequences, on the other hand, measure how long domination can be built through legal vertex choices. This paper introduces Grundy locating-hop domination sequences, in which a vertex choice is legal when it either footprints a previously undominated vertex through a closed hop neighborhood or strictly reduces an ambiguity potential that counts indistinguishable outside vertex pairs. The associated invariant is denoted by \( \gamma_{gr}^{\ell h}(G) \). General bounds are established, including \( \gamma_{\ell h}(G) \leq \gamma_{gr}^{\ell h}(G) \leq n(G) \). The parameter is additive over disjoint unions, and a hop-graph reduction identifies \( \gamma_{gr}^{\ell h}(G) \) with the corresponding locating-dominating sequence parameter on the hop graph \( G^{(2)} \). Exact values are obtained for complete graphs and stars. In particular, \( \gamma_{gr}^{\ell h}(K_n) = n \) and \( \gamma_{gr}^{\ell h}(K_{1,n}) = n \) for \( n \geq 2 \). Stars also give an infinite separation family: for \( n \geq 3 \), \( \gamma_{\ell h}(K_{1,n}) = 2 \) while \( \gamma_{gr}^{\ell h}(K_{1,n}) = n \).