Extremal problems of circle packings on a sphere and irreducible contact graphs

Recently, we have enumerated (up to isometry) all locally rigid packings of congruent circles (spherical caps) on the unit sphere with the number of circles $N<12$. This problem is equivalent to the enumeration of irreducible spherical contact graphs. In this paper, we show that using the list of irreducible contact graphs, one can solve various problems on extremal packings such as the Tammes problem for the sphere and projective plane, the problem of the maximum kissing number in spherical packings, Danzer's problems, and other problems on irreducible contact graphs.