Resolving dominating partitions in graphs

TL;DR

This paper investigates the properties of resolving and resolving dominating partitions in graphs, establishing bounds and characterizations for their minimum sizes, and providing Nordhaus-Gaddum bounds for these graph parameters.

Contribution

It introduces the concept of resolving dominating partitions, establishes bounds relating partition dimension and dominating partition dimension, and characterizes graphs with extremal values for these parameters.

Findings

Proves that ta_p(G) eta_p(G)+1.

Characterizes graphs with ta_p(G) and eta_p(G) near the order of the graph.

Provides tight Nordhaus-Gaddum bounds for eta_p(G) and ta_p(G).

Abstract

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 $\eta_p(G)$.