Triangle replacement for self-avoiding walks ($y^2=x^3+x^4$)

Replace every degree-three vertex of a graph in one class of a bipartite graph by a triangle with one port for each incident edge. A self-avoiding walk whose endpoints remain in the other class cannot make two completed passages through the same triangle: each passage needs two unused ports and only three exist. Each old two-edge passage has exactly two replacements, of lengths three and four. Therefore the restricted generating functions satisfy $Z_H^0(x)=Z_G^0(\sqrt{x^3+x^4})$. If the old radius is $1/\mu$, the new radius obeys ${\rho }^3+{\rho }^4={\mu }^{-2}$.