In the present research paper, we have studied the incomparability graph of the lattice $L^{2^{2}}_n$. In said graphs, we found a dominating set and the order of a graph. We have expressed the cardinality of neighbourhood of an atom making use of an expansion formula. We have also found the largest independent set of the mentioned graph.

Keywords: Incomparable pairs, dominating set, order of a graph, independent set, neighbourhood of an element.