  Using accurate dynamic polarizabilities of Li, Na, K and, Rb atoms, we
scrutinize the thermal Casimir-Polder interactions of these atoms with a single
layered graphene. Considering the modified Lifshitz theory for material
interactions, we reanalyze the dispersion coefficients ($C_3$s) of the above
atoms with graphene as functions of separation distance, gap parameter and
temperature among which some of them were earlier studied by estimating dynamic
polarizabilities of the above atoms from the single oscillator model
approximation. All these $C_3$ coefficients have been evaluated in the
framework of the Dirac model. The interactions are described for a wide range
of distances and temperatures to demonstrate the changes in behavior with the
varying conditions of the system and also sensitivities in the interactions are
analyzed by calculating them for different values of the gap parameter. From
these analyses, we find a suitable value of the gap parameter for which the
true nature of the interactions in graphene can be surmised more accurately.
