  The Grundy number of a graph $G$, denoted by $\Gamma(G)$, is the largest $k$
such that there exists a partition of $V(G)$, into $k$ independent sets
$V_1,\ldots, V_k$ and every vertex of $V_i$ is adjacent to at least one vertex
in $V_j$, for every $j < i$. The objects which are studied in this article are
families of $r$-regular graphs such that $\Gamma(G) = r + 1$. Using the notion
of independent module, a characterization of this family is given for $r=3$.
Moreover, we determine classes of graphs in this family, in particular the
class of $r$-regular graphs without induced $C_4$, for $r \le 4$. Furthermore,
our propositions imply results on partial Grundy number.
