  K-core and bootstrap percolation are widely studied models that have been
used to represent and understand diverse deactivation and activation processes
in natural and social systems. Since these models are considerably similar, it
has been suggested in recent years that they could be complementary. In this
manuscript we provide a rigorous analysis that shows that for any degree and
threshold distributions heterogeneous bootstrap percolation can be mapped into
heterogeneous k-core percolation and vice versa, if the functionality
thresholds in both processes satisfy a complementary relation. Another
interesting problem in bootstrap and k-core percolation is the fraction of
nodes belonging to their giant connected components $P_{\infty b}$ and
$P_{\infty c}$, respectively. We solve this problem analytically for arbitrary
randomly connected graphs and arbitrary threshold distributions, and we show
that $P_{\infty b}$ and $P_{\infty c}$ are not complementary. Our theoretical
results coincide with computer simulations in the limit of very large graphs.
In bootstrap percolation, we show that when using the branching theory to
compute the size of the giant component, we must consider two different types
of links, which are related to distinct spanning branches of active nodes.
