Constraint graph regularization

Use the degree-$128$ graphs $H_r$ of Expander size adjustment and laziness, whose unnormalized edge expansion is at least $h_0=7\mathrm{/}10$ when $r\ge 2$. For a constraint graph $G$ as in Constraint graph and labeling value with $E\mathrm{\ne }\mathrm{\varnothing }$, remove isolated vertices and replace each vertex of degree $r$ by a cloud of its $r$ incidence ports. Put $H_r$ inside that cloud, with equality on every edge. Keep one external edge for every original edge, joining its two designated ports and carrying its original relation. A loop's two ports are distinct. Call the resulting graph $G_1$.

On the $2\mathrm{\mid }E\mathrm{\mid }$ ports, add a copy of $H_{2\mathrm{\mid }E\mathrm{\mid }}$ with tautological relations, and at each port add $65$ ordinary tautological loops, i.e. $130$ loop slots. Call this $G_2$. Its degree is $129+128+130=387$; $G_1$ has degree $129$. The alphabet is unchanged. For an edgeless input, output the empty graph with value one; positive-degree and nonempty-size claims about $G_1,G_2$ are restricted to $E\mathrm{\ne }\mathrm{\varnothing }$. Fix an alphabet ordering for plurality tie breaking and for decoding removed isolated vertices.