Optimal vertex set partitions into different closed neighborhoods in powers of graphs and their complements

In a graph G = (V,E), a subset S subset of V is called an efficient dominating set if every vertex in V is dominated by exactly one vertex in S. A vertex v is an element of V is said to be dominated by a vertex in S if it either belongs to S or is adjacent to a vertex in S. Generalizing this notion, a k-efficient dominating set is defined via partitions of the vertex set into closed i-neighborhoods for 0 <= i <= k. The minimum cardinality of such a set is called the k-efficient domination number of G, denoted epsilon(k)(G). In this paper, we focus on determining epsilon(1)(G(k)), the 1-efficient domination number of the kth power of a graph G, and its complement Gk. We characterize graphs for which 1 <= epsilon(1)(G(k)) <= 3, and compute exact values of epsilon(1) for powers of paths, cycles, and their complements. Our results provide structural insights into neighborhood-based vertex partitions and efficient domination in graph powers.