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

For an ordered k-partition Ω = ₁, S₂, ..., Sₖ\ of vertex set of a connected graph G and a vertex v of G, the representation of v with respect to Ω is defined as the k-tuple r(v |Ω) = (d(v, S₁), d(v, S₂), ..., d(v, Sₖ )). The partition Ω is called a resolving partition of G, if r(u|Ω)≠ r(v|Ω) 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 γₖ(G). In this paper, we determine the partition dimension of cocktail party graph $CP(m+1)$ and corona product Gₘ̄. Moreover, we obtain k-domination numbers for $CP(m+1)$ and corona product Cₙₘ̄.