  We estimate the critical thresholds of bond and site percolation on
nonplanar, effectively two-dimensional graphs with chimera like topology. The
building blocks of these graphs are complete and symmetric bipartite subgraphs
of size $2n$, referred to as $K_{n,n}$ graphs. For the numerical simulations we
use an efficient union-find based algorithm and employ a finite-size scaling
analysis to obtain the critical properties for both bond and site percolation.
We report the respective percolation thresholds for different sizes of the
bipartite subgraph and verify that the associated universality class is that of
standard two-dimensional percolation. For the canonical chimera graph used in
the D-Wave Systems Inc.~quantum annealer ($n = 4$), we discuss device failure
in terms of network vulnerability, i.e., we determine the critical fraction of
qubits and couplers that can be absent due to random failures prior to losing
large-scale connectivity throughout the device.
