The Arrow Domination in Graphs

Document Type : Research Paper


1 Department of Mathematics, College of Education for Pure Sciences, University of Thi-Qar, Thi-Qar, Iraq

2 Department of‎ ‎Mathematics‎, ‎College of Education for Pure Sciences‎, ‎University of Thi-Qar‎, ‎Thi-Qar‎, ‎Iraq‎



‎The arrow domination is introduced in this paper with its inverse as a new type of domination‎. Let $G$ be a finite graph‎, ‎undirected‎, ‎simple and has no isolated vertex‎, ‎a set $D$ of $V(G)$ is said an arrow dominating set if $|N(w)\cap (V-D)|=i$ and $|N(w)\cap D|\geq j$ for every $w \in D$ such that $i$ and $j$ are two non-equal positive integers‎. ‎The arrow domination number $\gamma_{ar}(G)$ is the minimum cardinality over all arrow dominating sets in $G$‎. ‎Essential properties and bounds of arrow domination and its inverse when $i=1$ and $j=2$ are proved‎. ‎Then‎, ‎arrow domination number is discussed for several standard graphs and other graphs that formed by join and corona operations‎.