  In this paper the recently introduced concept of equidistant dimension
$eqdim(G)$ of graph $G$ is considered. Useful property of distance-equalizer
set of arbitrary graph $G$ has been established. For Johnson graphs $J_{n,2}$
and Kneser graphs $K_{n,2}$ exact values for $eqdim(J_{n,2})$ and
$eqdim(K_{n,2})$ have been derived, while for Johnson graphs $J_{n,3}$ it is
proved that $eqdim(J_{n,3}) \le n-2$. Finally, exact value of $eqdim(J_{2k,k})$
for odd $k$ has been presented.
