  Random K-out graphs, denoted $\mathbb{H}(n;K)$, are generated by each of the
$n$ nodes drawing $K$ out-edges towards $K$ distinct nodes selected uniformly
at random, and then ignoring the orientation of the arcs. Recently, random
K-out graphs have been used in applications as diverse as random (pairwise) key
predistribution in ad-hoc networks, anonymous message routing in
crypto-currency networks, and differentially-private federated averaging. In
many applications, connectivity of the random K-out graph when some of its
nodes are dishonest, have failed, or have been captured is of practical
interest. We provide a comprehensive set of results on the connectivity and
giant component size of $\mathbb{H}(n;K_n,\gamma_n)$, i.e., random K-out graph
when $\gamma_n$ of its nodes, selected uniformly at random, are deleted. First,
we derive conditions for $K_n$ and $n$ that ensure, with high probability
(whp), the connectivity of the remaining graph when the number of deleted nodes
is $\gamma_n=\Omega(n)$ and $\gamma_n=o(n)$, respectively. Next, we derive
conditions for $\mathbb{H}(n;K_n,\gamma_n)$ to have a giant component, i.e., a
connected subgraph with $\Omega(n)$ nodes, whp. This is also done for different
scalings of $\gamma_n$ and upper bounds are provided for the number of nodes
outside the giant component. Simulation results are presented to validate the
usefulness of the results in the finite node regime.
