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All Journal Electronic Journal of Graph Theory and Applications (EJGTA)
Nader Jafari Rad
Department of Mathematics, Shahrood University of Technology, Shahrood, Iran
Electrical & Electronics Engineering

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Bounds on weak and strong total domination in graphs M.H. Akhbari; Nader Jafari Rad
Electronic Journal of Graph Theory and Applications (EJGTA) Vol 4, No 1 (2016): Electronic Journal of Graph Theory and Applications
Publisher : GTA Research Group, Univ. Newcastle, Indonesian Combinatorics Society and ITB

Show Abstract | Download Original | Original Source | Check in Google Scholar | DOI: 10.5614/ejgta.2016.4.1.10

Abstract

A set $D$ of vertices in a graph $G=(V,E)$ is a total dominatingset if every vertex of $G$ is adjacent to some vertex in $D$. Atotal dominating set $D$ of $G$ is said to be weak if everyvertex $v\in V-D$ is adjacent to a vertex $u\in D$ such that$d_{G}(v)\geq d_{G}(u)$. The weak total domination number$\gamma_{wt}(G)$ of $G$ is the minimum cardinality of a weaktotal dominating set of $G$. A total dominating set $D$ of $G$ issaid to be strong if every vertex $v\in V-D$ is adjacent to avertex $u\in D$ such that $d_{G}(v)\leq d_{G}(u)$. The strongtotal domination number $\gamma_{st}(G)$ of $G$ is the minimumcardinality of a strong total dominating set of $G$. We presentsome bounds on weak and strong total domination number of a graph.
On open neighborhood locating-dominating in graphs Mustapha Chellali; Nader Jafari Rad; Suk Jai Seo; Peter James Slater
Electronic Journal of Graph Theory and Applications (EJGTA) Vol 2, No 2 (2014): Electronic Journal of Graph Theory and Applications
Publisher : GTA Research Group, Univ. Newcastle, Indonesian Combinatorics Society and ITB

Show Abstract | Download Original | Original Source | Check in Google Scholar | DOI: 10.5614/ejgta.2014.2.2.1

Abstract

A set D of vertices in a graph G = (V (G), E(G)) is an open neighborhood locating-dominating set (OLD-set) for G if for every two vertices u, v of V (G) the sets N(u) ∩ D and N(v) ∩ D are non-empty and different. The open neighborhood locating-dominating number OLD(G) is the minimum cardinality of an OLD-set for G. In this paper we characterize graphs G of order n with OLD(G) = 2, 3, or n and graphs with minimum degree (G) ≥ 2 that are C4-free with OLD(G) = n-1.
Co-Authors M.H. Akhbari Mustapha Chellali Peter James Slater Suk Jai Seo