The undirected underlying graph of a 2-quasi best match graph (2-qBMG) is proven not to contain any induced graph isomorphic to $P_6$ or $C_6$. This new feature allows for the investigation of 2-BMGs further by exploiting the numerous known results on $P_6$ and $C_6$ free graphs together with the available polynomial algorithms developed for their studies. In this direction, there are also some new contributions about dominating bicliques and certain vertex decompositions of the undirected underlying graph of a 2-qBMG.