I was thinking of first a comparison of all the stars that make a circuit, and then an algorithm that first gets you in that circuit and then finds the closest star in that circuit.

For example stars K1, K2, K3, and K4 make a circuit and you can go around in a circle with them. There's another circuit with K2, K3, and the star off to the side. Those two stray stars would be considered a circuit even though there is 1, but both are their own circuits.

First step is to get yourself in the circuit the star is in, but the starting location is in the circuit which contains K4. Then, you've got to travel either left or right, and check the total distance of all the lines from the left and right, and in this case the left path is shorter (directly).

I'm not sure if this is a good algorithm or not, but I would write a function to run through and recursively catalog all the stars and which circuits they make up. For faster speeds, this would run at compilation.