On independent domination numbers of grid and toroidal grid directed graphs

• Main Author: R. Shaheen
• Format: Article
• Language: English
• Published: Azarbaijan Shahide Madani University 2019-06-01
• Series: Communications in Combinatorics and Optimization
• Subjects: directed path; directed cycle; cartesian product; independent domination number
• Online Access: http://comb-opt.azaruniv.ac.ir/article_13846_0948ec1c34ebfc23d9e6b9f6dc3f735d.pdf

A subset $S$ of vertex set $V(D)$ is an independent dominating set of a digraph $D$ if $S$ is both an independent and a dominating set of $D$. The independent domination number $i(D)$ is the minimum cardinality of an independent dominating set of $D$. In this paper we calculate the independent domination number of the Cartesian product of two directed paths $P_m$ and $P_n$ for arbitraries $m$ and $n$. Also, we determine the independent domination number of the Cartesian product of two directed cycles $C_m$ and $C_n$ for $m, n \equiv 0 \pmod 3$ and $n \equiv 0\pmod m$. We note that, there are many values of $m$ and $n$ such that $C_m \Box C_n$ does not have an independent dominating set.