  The topological vertex formalism for 5d $\mathcal{N}=1$ gauge theories is not
only a convenient tool to compute the instanton partition function of these
theories, but it is also accompanied by a nice algebraic structure that reveals
various kinds of nice properties such as dualities and integrability of the
underlying theories. The usual refined topological vertex formalism is derived
for gauge theories with $A$-type quiver structure (and $A$-type gauge groups).
In this article, we propose a construction with a web of vertex operators for
all $ABCDEFG$-type and affine quivers by introducing several new vertices into
the formalism, based on the reproducing of known instanton partition functions
and qq-characters for these theories.
