Resolving dominating partitions in graphs

A partition $\Pi=\{S_1,\ldots,S_k\}$ of the vertex set of a connected graph $G$ is called a \emph{resolving partition} of $G$ if for every pair of vertices $u$ and $v$, $d(u,S_j)\neq d(v,S_j)$, for some part $S_j$. The \emph{partition dimension} $\beta_p(G)$ is the minimum cardinality of a resolving partition of $G$. A resolving partition $\Pi$ is called \emph{resolving dominating} if for every vertex $v$ of $G$, $d(v,S_j)=1$, for some part $S_j$ of $\Pi$. The \emph{dominating partition dimension} $\eta_p(G)$ is the minimum cardinality of a resolving dominating partition of $G$. In this paper we show, among other results, that $\beta_p(G) \le \eta_p(G) \le \beta_p(G)+1$. We also characterize all connected graphs of order $n\ge7$ satisfying any of the following conditions: $\eta_p(G)= n$, $\eta_p(G)= n-1$, $\eta_p(G)= n-2$ and $\beta_p(G) = n-2$. Finally, we present some tight Nordhaus-Gaddum bounds for both the partition dimension $\beta_p(G)$ and the dominating partition dimension $\eta_p(G)$.