The domination number of the king’s graph

Abstract

For a graph $\Gamma$, a subset $S\subseteq V_\Gamma$ is known to be a dominating set, if every $x\in V_\Gamma {\setminus } S$ has at least one neighbor in D. The domination number $\gamma (\Gamma )$ is merely the size of a smallest dominating set in $\Gamma$. The strong product $P_r\boxtimes P_s$ of two paths $P_r$ and $P_s$ is known as the king’s graph. Interestingly, the king’s graph is isomorphic to the two-parametric family of cellular neural network (CNNs). In Asad et al. (Alex Eng J 66:957–977, 2023), the authors retrieved certain structural characteristics of CNNs from their minimal dominating sets. They conjectured in Problem 8.3 that the domination number of $P_r\boxtimes P_s$ is $\Big \lceil \frac{r}{3}\Big \rceil \Big \lceil \frac{s}{3}\Big \rceil$. This paper solves Problem 8.3 by providing a proof to the conjecture. This result, in turn, reveals interesting topological properties such as an optimal routing for this class of neural networks.