Insertion algorithms for Type $\mathrm{B}$ and $\mathrm{D}$ Gelfand $W$-graphs

Like the RSK correspondence for symmetric groups, Garfinkle defined a domino correspondence for type $\mathrm{B}$ and $\mathrm{D}$ Coxeter groups. Similar to the Knuth relations, Taskin and Pietraho give the plactic relations for the domino correspondence and Bonnafé use them to classify the cells for type $\mathrm{B}$ Coxeter groups. We give some further properties of the plactic relations and use these relations to describe the bidirected edges and the molecules of Gelfand $W$-graphs for type $\mathrm{B}$ and $\mathrm{D}$ Coxeter groups.