  Recently, Gaiotto and Rapcak (GR) proposed a new family of the vertex
operator algebra (VOA) as the symmetry appearing at an intersection of
five-branes to which they refer as Y algebra. Prochazka and Rapcak, then
proposed to interpret Y algebra as a truncation of affine Yangian whose module
is directly connected to plane partitions (PP). They also developed GR's idea
to generate a new VOA by connecting plane partitions through an infinite leg
shared by them and referred it as the web of W-algebra (WoW). In this paper, we
demonstrate that double truncation of PP gives the minimal models of such VOAs.
For a single PP, it generates all the minimal model irreducible representations
of W-algebra. We find that the rule connecting two PPs is more involved than
those in the literature when the U(1) charge connecting two PPs is negative.
For the simplest nontrivial WoW, N=2 superconformal algebra, we demonstrate
that the improved rule precisely reproduces the known character of the minimal
models.
