Stratified Claw Domination in Prisms

A graph $G$ is $2$-stratified if its vertex set is partitioned into two classes (each of which is a stratum or a color class), where the vertices in one class are colored red and those in the other class are colored blue. Let $F$ be a $2$-stratified graph rooted at some blue vertex $v$. An $F$-coloring of a graph is a red-blue coloring of the vertices of $G$ in which every blue vertex $v$ belongs to a copy of $F$ rooted at $v$. The $F$-domination number $\gamma_F(G)$ is the minimum number of red vertices in an $F$-coloring of $G$. In this paper we determine the $F$-domination number of the prisms $C_n \times K_2$ for all 2-stratified claws $F$ rooted at a blue vertex.