Distance k-domination and k-resolving domination of the corona product of graphs

For two simple graphs G and H, the vertex corona product of G and H, denoted by G ⊙ H, is the graph obtained by adding a copy of H for each vertex of G and joining each vertex of G to all vertices in 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 a vertex in D. The minimum cardinality overall distance k-dominating sets of G is the distance k-domination number, denoted by γk(G). The metric dimension of a graph is the smallest number of vertices required to distinguish all other vertices based on distances uniquely. The concept of distance k-resolving domination in graphs combines both 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 vertex corona product of graphs. First, we show that for k ≥ 2, the distance k-domination number of G ⊙ H is equal to γk−1(G) for any two graphs G and H. Then, we give the exact value of γk(G ⊙ H) when G is a complete graph, complete m-partite graph, path and cycle. We also provide general bounds for γk(G ⊙ H). Then, we examine the distance k-resolving domination number for G ⊙ H. For k = 1, we give bounds for γ ^r (G ⊙ H) the resolving domination number of G ⊙ H and characterize the graphs achieving those bounds. Later, for k ≥ 2, we establish bounds for γ^r _k (G ⊙ H) the distance k-resolving domination number of G ⊙ H and characterize the graphs achieving these bounds.