  Let $\Delta$ be the Artin complex of the Artin group of type $D_n$. This
complex is also called the spherical Deligne complex of type $D_n$. We show
certain types of 6-cycles in the 1-skeleton of $\Delta$ either have a center,
which is a vertex adjacent to each vertex of the 6-cycle, or a quasi-center,
which is a vertex adjacent to three of the alternating vertices of the 6-cycle.
This will be a key ingredient in proving $K(\pi,1)$-conjecture for several
classes of Artin groups in a companion article.
  As a consequence, we also deduce that certain 2-dimensional relative Artin
complex inside the $D_n$-type Artin complex, endowed with the induced Moussong
metric, is CAT$(1)$.
