Rhombus Grid

Relevant wiki: Rectangular Grid Walk

We can obtain the number of paths easily by using this method:

Give every dot a value of $0$ .
Add $1$ to $A$ :s value.
Add $A$ :s value to the dots to which can be moved from $A$ .
Repeat the last step for every new dot you add to until you get to $B$ .
The number of paths is the value of $B$ .

Let's divide the problems into $3$ separate cases.

Case 1: Passing trough no stars
Imagine that the dots with a star don't exist and calculate the number of possible paths which is $6$ .

Case 2: Passing trough the first star
Imagine that the dots above and below the first star don't exist (forcing the path go through the first star) and calculate the number of possible paths which is $12$ .

Case 3: Passing trough the last star
By symmetry the number of paths is also $12$ .

Adding the values we obtain the answer: $6 + 12 + 12 = \boxed{30}$ .

You could also calculate the number of all paths and then subtract the number of paths which go through both stars. This also gives the same answer

Just a matter of waiting out rounds in the roster. If you count them with passing through the first star, you May double your answer because of the symmetry. So op the outside you got 1's. Gettin to the star makes 2, from the star to outside makes 3 routes above and 3 routes down. You got 6 routes to middle. Since you can't pass trough the 2nd star, there are 9 rounds around via the top and 9 via the downside. Hence 18 routes from A to B through the first star. This makes 36 routes in total.