Number of rings not found and that "potentially exist"

One of the first information it is possible to extract from ring statistics, except the number of rings, is the number of rings not found by the analysis. Indeed calculation times do strongly depend on the maximum search depth, ie. the maximum size of a ring. To carry out the analysis this value has to be chosen to get the best possible compromise between CPU time and quality of the description.
Nevertheless whatever this limiting value is, some rings of a size bigger than the maximum search depth may not be found by the analysis. In the King and the Guttman's criteria it is possible to evaluate the number of "potentially not found" rings or rings that "potentially exist".
Thus for a given atom At we can consider that a closed path exists and is not found:

  1. If the atom At has at least 2 nearest neighbors

  2. If no closed path is found:

    1. Starting from one neighbor to go back on the considered atom At (Guttman's criterion)

    2. Between one couple of neighbors of the atom At (King's criterion)

  3. If the 2 nearest neighbors of the atom At have at least 2 nearest neighbors (to avoid non bridging atoms)

Thus if during the analysis these 3 conditions are full filled (1, 2-a, 3 for the Guttman's criterion, and 1, 2-b, 3 for the King's criterion) then we can say that this analysis has potentially missed a ring between the neighbors of atom At. The smaller this number of "potentially" missed rings will be the better this analysis will be and the better the description of the connectivity of the material studied will be. The term "potentially" has been chosen because the method only allows to avoid first neighbor non bridging atoms.