  Let $\Gamma=(X,\mathcal{R})$ denote a finite, simple, connected, and
undirected non-bipartite graph with vertex set $X$ and edge set $\mathcal{R}$.
Fix a vertex $x \in X$, and define $\mathcal{R}_f = \mathcal{R} \setminus \{yz
\mid \partial(x,y) = \partial(x,z)\}$, where $\partial$ denotes the path-length
distance in $\Gamma$. Observe that the graph $\Gamma_f=(X,\mathcal{R}_f)$ is
bipartite. We say that $\Gamma$ supports a uniform structure with respect to
$x$ whenever $\Gamma_f$ has a uniform structure with respect to $x$.
  Assume that $\Gamma$ is a distance-regular graph with classical parameters
$(D,q,\alpha,\beta)$ with $q \le 1$. Recall that $q$ is an integer, which is
not equal to $0$ or $-1$. The purpose of this paper is to study when $\Gamma$
supports a uniform structure with respect to $x$. The main result of the paper
is a complete classification of graphs with classical parameters with $q\leq 1$
and $D \ge 4$ that support a uniform structure with respect to $x$.
