The Tammes problem is a problem in packing a given number N of equal circles on the surface of a sphere with their common radius as large as possible. The Tammes problem is presently solved only for several values of N: for N=3,4,6,12 by L. Fejes Toth (1943); for N=5,7,8,9 by Schutte and van der Waerden (1951); for N=10,11 by Danzer (1963); and for N=24 by Robinson (1961).

Recently, I and Alexey Tarasov solved the Tammes problem for N=13. This computer-assisted proof is based on an enumeration of maximal irreducible contact graphs with 13 vertices. Relying on these ideas, now we are working on properties of irreducible graphs, devoting special attention to maximal irreducible graphs for the Tammes and related problems.