  We investigate the statistics of articulation points and bredges
(bridge-edges) in complex networks in which bonds are randomly removed in a
percolation process. Articulation points are nodes in a network which, if
removed, would split the network component on which they are located into two
or more separate components, while bredges are edges whose removal would split
the network component on which they are located into two separate components.
Both articulation points and bredges play an important role in processes of
network dismantling and it is therefore useful to know the evolution of the
probability of nodes or edges to be articulation points and bredges,
respectively, when a fraction of edges is randomly removed from the network in
a percolation process. Due to the heterogeneity of the network, the probability
of a node to be an articulation point, or the probability of an edge to be a
bredge will not be homogeneous across the network. We therefore analyze full
distributions of articulation point probabilities as well as bredge
probabilities, using a message-passing or cavity approach to the problem, as
well as a deconvolution of these distributions according to degrees of the node
or the degrees of both adjacent nodes in the case of bredges. Our methods allow
us to obtain these distributions both for large single instances of networks as
well as for ensembles of networks in the configuration model class in the
thermodynamic limit of infinite system size. We also derive closed form
expressions for the large mean degree limit of Erd\H{o}s-R\'enyi networks.
