Formulas for the Number of Weak Homomorphisms from Paths to Ladder Graphs and Stacked Prism Graphs

Let $G$ and $H$ be graphs. A mapping $f$ from $V\left(G\right)$ to $V\left(H\right)$ is called a weak homomorphism from $G$ to $H$ if $f\left(x\right)=f\left(y\right)$ or $\left\{f\left(x\right),f\left(y\right)\right\}\in E\left(H\right)$ whenever $\left\{x,y\right\}\in E\left(G\right)$ . A ladder graph is the Cartesian product of two paths, where one of the paths has only one edge. A stacked prism graph is the Cartesian product of a path and a cycle. In this paper, we provide a formula to determine the number of weak homomorphisms from paths to ladder graphs and a formula to determine the number of weak homomorphisms from paths to stacked prism graphs.