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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.