Distance k-domination and k-resolving domination of the corona product of graphs - Retnowardani, Dwi Agustin; Susilowati, Liliek; Dafik; Dliou, Kamal (Statistics, Optimization & Information Computing, 2024)
For two simple graphs G and H, the corona product of G and H is the graph obtained by adding a copy of H for every vertex of G and joining each vertex of G to its corresponding copy of H. For k ≥ 1, a set of vertices D in a graph G is a distance k-dominating set if any vertex in G is at a distance less or equal to k from some vertex in D. The minimum cardinality overall distance k-dominating sets of G is the distance k-domination number, denoted by γ_k(G). The metric dimension of a graph is the smallest number of vertices required to distinguish all other vertices based on distances uniquely. The distance k-resolving domination in graphs combines distance k-domination and the metric dimension of graphs. In this paper, we investigate for all k≥ 1, the distance k-domination and the distance k-resolving domination in the corona product of graphs. First, we show that for k≥ 2 the distance k-domination number of GH is equal to γ_k-1(G) for any two graphs G and H. Then, we give the exact value of γ_k(GH) when G is a complete graph, complete m-partite graph, path and cycle. We also provide general bounds for γ_k(GH). Then, we examine the distance k-resolving domination number for GH. For k=1, we give bounds for γ^r(GH) the resolving domination number of GH and characterize the graphs achieving those bounds. Later, for k≥ 2, we establish bounds for γ^r_k(GH) the distance k-resolving domination number of GH and characterize the graphs achieving these bounds.