Weak total resolving sets in graphs
  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. 
Imran Javaid, Muhammad Salman, Mahr Murtaza, Farheen Iftikhar,   Muhammad Imran
math.CO