We begin by exploring the idea of complementation in various areas of life and mathematics. We then focus on properties of a graph and investigate whether these properties are preserved in the graph’s complement. For example, can we expect the existence of a planar graph on 8 vertices whose complement is also planar? What about such a graph on 9 vertices? Does there exist a graph of order 6 whose complement is isomorphic to the graph itself? We examine several properties, including linkless embeddability and knotless embeddability.