  This paper describes a search for events with one top-quark and large missing
transverse momentum in the final state. Data collected during 2015 and 2016 by
the ATLAS experiment from 13 TeV proton-proton collisions at the LHC
corresponding to an integrated luminosity of 36.1 fb$^{-1}$ are used. Two
channels are considered, depending on the leptonic or the hadronic decays of
the $W$ boson from the top quark. The obtained results are interpreted in the
context of simplified models for dark-matter production and for the single
production of a vector-like $T$ quark. In the absence of significant deviations
from the Standard Model background expectation, 95% confidence-level upper
limits on the corresponding production cross-sections are obtained and these
limits are translated into constraints on the parameter space of the models
considered.

High Energy Physics - Experiment