  In the framework of spin-0 $s$-channel simplified models, we explore the
possibility of assessing the structure of dark matter interactions through the
associate production of dark matter and $t\bar{t}$ pairs. To this purpose,
final states with two leptons are considered and the kinematic properties of
the dilepton system is studied. We develop a realistic analysis strategy and
provide a detailed evaluation of the achievable sensitivity for the dark matter
signal assuming integrated luminosities of $300 \, {\rm fb}^{-1}$ and $3 \,
{\rm ab}^{-1}$ at the 14 TeV LHC. Furthermore, upper limits on the mediator
masses for which the two different CP hypotheses can be distinguished are
derived. The obtained limits on the signal strengths are finally translated
into constraints on the parameter space of two spin-0 simplified models
including a scenario with an extended Higgs sector.
