  The work of C. Bonnaf{\'e}, M. Geck, L. Iancu and T. Lam \cite{Geck-Lam}
shows through two conjectures that $r$-domino tableaux have an important role
in Kazhdan-Lusztig theory of type $B$ with unequal parameters. In this paper we
provide plactic relations on signed permutations which determine whether given
two signed permutations have the same insertion $r$-domino tableaux in
Garfinkle's algorithm \cite{Garfinkle1}.
