  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 $\nu_B$ of the backbone in the net-like
phase; graph dimensions -- $d_{\min}$ of the tree-like phase, and $D_{\min}$ of
the backbone in the net-like phase appear to be universal within the accuracy
of our calculations, while the backbone fractal dimension $D_B$ is not
universal: it depends on the parameter of a model.
