We numerically study one-parameter family of random single-cluster systems. A
finite-concentration topological phase transition from the net-like to the
tree-like phase (the latter is without a backbone) is present in all models of
the class. Correlation radius index νB of the backbone in the net-like
phase; graph dimensions -- dmin of the tree-like phase, and Dmin of
the backbone in the net-like phase appear to be universal within the accuracy
of our calculations, while the backbone fractal dimension DB is not
universal: it depends on the parameter of a model.Comment: Published variant; more accurate numerical data and minor
corrections. 4 pages, 5 figure