THE PARTITION DIMENSION AND k-DOMINATION NUMBER OF TWO SPECIFIC GRAPHS

For an ordered k-partition = \S₁, S₂,. . . , Sₖ\ of vertex set of a connected graph G and a vertex v of G, the representation of v with respect to is defined as the k-tuple r (v |) = (d (v, S₁), d (v, S₂),. . . , d (v, Sₖ) ). The partition is called a resolving partition of G, if r (u|) r (v|) for all distinct u, v V (G). The partition dimension of a graph G, denoted by pd (G), is the cardinality of a minimum resolving partition of G. A subset D V (G) is k-dominating in G, if every vertex of V (G) D has at least k neighbors in D. The minimum cardinality among all k-dominating sets is called the k-domination number of G, denoted by ₖ (G). In this paper, we determine the partition dimension of cocktail party graph CP (m+1) and corona product Gₘ. Moreover, we obtain k-domination numbers for CP (m+1) and corona product Cₙₘ.