You are again given a constraint network that encodes a graph coloring problem. The graph is shown in the picture below.

The network is formally specified as follows:

${\cal C} = \{V, dom, (R_{uv})\}$ with variables $V = \{v_1, \dots, v_7\}$ and domains $dom(v_3) = dom(v_4) = dom(v_7) = \{b,r\}$, $dom(v_2) = dom(v_5) = \{b, g\}$, $dom(v_1) = \{b,g,r\}$, and $dom(v_6) = \{g,r,y\}$.

The constraints are such that no two neighboring nodes in the graph can be colored in the same color, i.e., there is a constraint $v_i \neq v_j$ for all pairs $v_i, v_j \in V$ if $i \neq j$ and there exists an edge in the graph connecting $v_i$ and $v_j$.

Including inconsistent ones, the network has 288 assignments.

Assume you are given a partial assignment $\alpha = \{v_1 \mapsto b\}$, perform forward checking.

How many assignments does the network have after forward checking when $v_1$ is assigned to $b$?

You are given the same constraint network as in the previous exercise. Again, we assigned $v_1$ the value $b$.

How many assignments does the network have after performing $AC{-}1$ on the partial assignment $\alpha = \{v_1 \mapsto b\}$?