February 9, 2023

# Doost Ali Mojdeh

Degree: Professor
Address: Department of Mathematics, University of Mazandaran, Babolsar, Iran
Education: Ph.D in Mathematics (Graph Theory and Combinatorics)
Phone: 011-35302448
Faculty: Faculty of Mathematical Sciences

## Research

 Title ON LOCATING AND DIFFERENTIATING- TOTAL K-DOMINATING SET IN GRAPHS Type Article Keywords Total k-dominating set; locating-total k-dominating set; differentiating-total k-dominating set; graph; tree. Journal Asian Journal of Mathematics and Computer Research DOI — Researchers a { color: #4f98b0; } a:hover { color: #ffab00; } a:link:visited { text-decoration: none; } Doost Ali Mojdeh (First researcher) , Ashraf Seyed Khalkhali (Second researcher)

## Abstract

Let $G$ be a graph with vertex set $V=V(G)$ and edge set $E=E(G)$. A subset $S$ of $V$ is a total $k$-dominating set of $G$ if every vertex in $V$ is within distance $k$ from some vertex in $S$. The total $k$-domination number, $\gamma_{t}^{k}(G)$ is the minimum cardinality of a total $k$-dominating set of $G$. We are interested in to introduce two types of total $k$-dominating sets, namely locating-total $k$-dominating set of a graph $G$ ($\gamma_{t,k}^{L}(G)$-set) and differentiating-total $k$-dominating set of a graph $G$ ($\gamma_{t,k}^{D}(G)$-set). We show that; for any graph $G$ of order $n$, $\log_2 {n}-1\leq \gamma_{t,k}^{L}(G)\leq \frac{3}{5}n$. We obtain sharp upper and lower bounds for $\gamma_{t,k}^{L}(T)$ and $\gamma_{t,k}^{D}(T)$ for every tree $T$.