Two-Fold Circle-Covering of the Plane under Congruent Voronoi Polygon Conditions

The $k$-coverage problem is to find the minimum number of disks such that each point in a given plane is covered by at least $k$ disks. Under unit disk condition, when $k$=1, this problem has been solved by Kershner in 1939. However, when $k > 1$, it becomes extremely difficult. One tried to tackle this problem with different restrictions. In this paper, we restrict ourself to congruent Voronoi polygon, and prove the minimum density of the two-coverage with such a restriction. Our proof is simpler and more rigorous than that given recently by Yun et al.