Relations between Metric Dimension and Domination Number of Graphs

TL;DR

This paper explores the relationship between the metric dimension and domination number of graphs, establishing bounds and characterizing cases of equality, with implications for understanding graph structure.

Contribution

It proves a new upper bound for the metric dimension based on the domination number and characterizes graphs where equality holds.

Findings

Established that eta(G)  n - (G) for connected graphs.

Equality holds iff G is complete or a complete bipartite graph with parts of size at least 2.

Derived new bounds for eta(G) using degree parameters.

Abstract

A set $W\subseteq V(G)$ is called a resolving set, if for each two distinct vertices $u,v\in V(G)$ there exists $w\in W$ such that $d(u,w)\neq d(v,w)$, where $d(x,y)$ is the distance between the vertices $x$ and $y$. The minimum cardinality of a resolving set for $G$ is called the metric dimension of $G$, and denoted by $\beta(G)$. In this paper, we prove that in a connected graph $G$ of order $n$, $\beta(G)\leq n-\gamma(G)$, where $\gamma(G)$ is the domination number of $G$, and the equality holds if and only if $G$ is a complete graph or a complete bipartite graph $K_{s,t}$, $ s,t\geq 2$. Then, we obtain new bounds for $\beta(G)$ in terms of minimum and maximum degree of $G$.