  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.
