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

For an ordered k-partition \Omega = \{S_۱, S_۲, ..., S_k\} of vertex set of a connected graph G and a vertex v of G, the representation of v with respect to \Omega is defined as the k-tuple r(v |\Omega) = (d(v, S_۱), d(v, S_۲), ..., d(v, S_k )). The partition \Omega is called a resolving partition of G, if r(u|\Omega)\neq r(v|\Omega) for all distinct u, v \in 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\subseteq V(G) is k-dominating in G, if every vertex of V(G)\setminus 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 \gamma_k(G). In this paper, we determine the partition dimension of cocktail party graph CP(m+۱) and corona product G\circ\overline{K_m}. Moreover, we obtain k-domination numbers for CP(m+۱) and corona product C_n\circ\overline{K_m}.

کلیدواژه ها:

Resolving set ، partition dimension ، domination number ، cocktail party graph ، corona product