Weak total resolving sets in graphs

TL;DR

This paper introduces and studies the concept of weak total resolving sets and weak total metric dimension in graphs, focusing on trees and exploring properties, characterizations, and random models.

Contribution

It defines weak total resolving sets and metric dimension, investigates these in trees, and explores their properties, characterizations, and realizations in graphs.

Findings

Weak total metric dimension of trees is characterized.

Weak total resolving number and related concepts are introduced and studied.

Some graphs' weak total resolving numbers and metric dimensions are characterized and realized.

Abstract

A set $W$ of vertices of $G$ is said to be a weak total resolving set for $G$ if $W$ is a resolving set for $G$ as well as for each $w\in W$, there is at least one element in $W-\{w\}$ that resolves $w$ and $v$ for every $v\in V(G)- W$. Weak total metric dimension of $G$ is the smallest order of a weak total resolving set for $G$. This paper includes the investigation of weak total metric dimension of trees. Also, weak total resolving number of a graph as well as randomly weak total $k$-dimensional graphs are defined and studied in this paper. Moreover, some characterizations and realizations regarding weak total resolving number and weak total metric dimension are given.