  Graphite is one of the many presolar circumstellar condensate species found
in primitive meteorites. While the isotopic compositions of low-density
graphite grains indicate an origin in core-collapse supernovae, some
high-density grains have extreme isotopic anomalies in C, Ca and Ti, which
cannot be explained by envelope predictions of asymptotic giant branch (AGB)
stars or theoretical supernova models. The Ca and Ti isotopic anomalies,
however, match the predictions of He-shell abundances in AGB stars. In this
study, we show that the C, Ca, and Ti isotopic anomalies are consistent with
nucleosynthesis predictions of the H-ingestion phase during a very late thermal
pulse (VLTP) event in post-AGB stars. The low $^{12}$C/$^{13}$C isotopic ratios
in these grains are a result of abundant $^{12}$C efficiently capturing the
protons that are being ingested during the VLTP. Very high neutron densities of
$\sim 10^{15}$ cm$^{-3}$, typical of the $i$-process, are achieved during this
phase in post-AGB stars. The large $^{42,43,44}$Ca excesses in some graphite
grains are indicative of neutron capture nucleosynthesis during VLTP. The
comparison of VLTP nucleosynthesis calculations to the graphite data also
indicate that apparent anomalies in the Ti isotopic ratios are due to large
contributions from $^{46,48}$Ca, which cannot be resolved from the isobars
$^{46,48}$Ti during the measurements. We conclude that presolar graphite grains
with moderate to extreme Ca and Ti isotopic anomalies originate in post-AGB
stars that suffer a very late thermal pulse.
