Suppose we have a graph $G(V,E)$ and we can colour some vertex's in black.

Colouring is called tolerant if no two joined (by edge) vertex's are coloured both.

Is there algorithm or formula of estimating number of all possible tolerant colourings of certain graph?

Actualy i need to do it for such graph

[![enter image description here][1]][1]


  [1]: https://i.sstatic.net/u1bIH.png

I have an idea to use reccurence
$$|G|=|G\backslash \{v\}|+|G\backslash n(v)|$$
where $|G|$ is the number all possible tollerant colourings of graph G,
$v$ is an arbitrary vertex, and $n(v)$ is a set of all vertex's joined to $v$ and $v$ itself.

But may be there some better ideas or results?