  The random-cluster model, a correlated bond percolation model, unifies a
range of important models of statistical mechanics in one description,
including independent bond percolation, the Potts model and uniform spanning
trees. By introducing a classification of edges based on their relevance to the
connectivity we study the stability of clusters in this model. We derive
several exact relations for general graphs that allow us to derive
unambiguously the finite-size scaling behavior of the density of bridges and
non-bridges. For percolation, we are also able to characterize the point for
which clusters become maximally fragile and show that it is connected to the
concept of the bridge load. Combining our exact treatment with further results
from conformal field theory, we uncover a surprising behavior of the variance
of the number of (non-)bridges, showing that these diverge in two dimensions
below the value $4\cos^2{(\pi/\sqrt{3})}=0.2315891\cdots$ of the cluster
coupling $q$. Finally, it is shown that a partial or complete pruning of
bridges from clusters enables estimates of the backbone fractal dimension that
are much less encumbered by finite-size corrections than more conventional
approaches.
