International Journal of Nonlinear Analysis and Applications
The arrow domination in graphs
Document Type : Research Paper
Authors
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
Abstract
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.
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