  We construct novel web diagrams with a trivalent or quadrivalent gluing for
various 6d/5d theories from certain Higgsings of 6d conformal matter theories
on a circle. The theories realized on the web diagrams include 5d Kaluza-Klein
theories from circle compactifications of the 6d $G_2$ gauge theory with 4
flavors, the 6d $F_4$ gauge theory with 3 flavors, the 6d $E_6$ gauge theory
with 4 flavors and the 6d $E_7$ gauge theory with 3 flavors. The Higgsings also
give rise to 5d Kaluza-Klein theories from twisted compactifications of 6d
theories including the 5d pure SU(3) gauge theory with the Chern-Simons level 9
and the 5d pure SU(4) gauge theory with the Chern-Simons level 8. We also
compute the Nekrasov partition functions of the theories by applying the
topological vertex formalism to the newly obtained web diagrams.
