Let the distance between the two given vertices be 1. If the two given vertices are adjacent, then the other vertices lie on four circles, two of radius 1 and two of radius $\sqrt{2}$. If the two vertices are separated by a diagonal of a face of the cube, then the locus of possible vertices adjacent to both of them is a circle of radius $\frac{1}{2}$, the locus of possible vertices adjacent to exactly one of them is two circles of radius $\frac{\sqrt{2}}{2}$, and the locus of possible vertices adjacent to neither of them is a circle of radius $\frac{\sqrt{3}}{2}$. If the two given vertices are separated by a long diagonal, then each of the other vertices lie on one of two circles of radius $\frac{\sqrt{2}}{3}$, for a total of 10 circles.