E overlap quotients terminate at a pure blockade

The E overlap blockade and mixed quotient sequence supplies the following definition: Fix a comb block $B_i$ with nonempty $E$ overlap support $X_i$. By lem-e-overlap-classes-form-an-anticonnected-partition, its overlap classes are nonempty anticonnected sets partitioning $X_i$. Fix an enumeration of the finite set $B_i$, and order the classes by their least enumerated vertex to obtain ${\mathcal{L}}^1$. Define recursively ${\mathcal{L}}^{s+1}={\mathcal{L}}^s\mathrm{/}\mathcal{M}$ for $s\ge 1$, using def-quotient-blockade-by-mixed-block-reachability and its least-member ordering. Thus one replaces each mixed-reachability class of blocks by its union. This construction is used only when $X_i\mathrm{\ne }\mathrm{\varnothing }$.

A quotient block of connected or anticonnected blocks is again connected or anticonnected supplies the following statement: Let $\mathcal{L}$ be a blockade and let $D$ be a block of the quotient blockade $\mathcal{L}\mathrm{/}\mathcal{M}$. 1. If every block of $\mathcal{L}$ contained in $D$ induces a connected subgraph, then $G[D]$ is connected. 2. If every block of $\mathcal{L}$ contained in $D$ induces an anticonnected subgraph, then $G[D]$ is anticonnected.

The well-ordering principle supplies the following statement: Every nonempty subset $S\subseteq \mathbb{N}$ has a least element: there is $\mathrm{\ell }\in S$ with $\mathrm{\ell }\le s$ for all $s\in S$.