The terminal E overlap pattern is E-free

E overlap quotients terminate at a pure blockade supplies the following statement: For nonempty overlap support, put $n=\mathrm{\mid }{\mathcal{L}}^1\mathrm{\mid }$. Every stage ${\mathcal{L}}^s$ partitions the same support into nonempty anticonnected blocks and coarsens ${\mathcal{L}}^1$. There is a least $q\ge 1$ for which ${\mathcal{L}}^q$ is pure, with $q\le n$. At most $n-1$ strict transitions occur, and all stages from $q$ onward are identical.

E overlap chains inside one comb block supplies the following definition: Fix a block $B_i$ of a finite graph comb (def-comb-in-a-graph). Let ${\mathcal{E}}_i$ consist of all six-vertex subsets of $B_i$ inducing the graph in def-e-graph-and-co-e-graph. Put $X_i={\bigcup }_{S\in {\mathcal{E}}_i}S$ and $Y_i=B_i\setminus X_i$. For $d,e\in X_i$, define $d<msub><mi>R</mi><mi>i</mi></msub>e$ if there exist $m\ge 0$ and vertices $d=d_0,d_1,\dots ,d_m=e$ in $X_i$ such that each consecutive pair is contained in some $S\in {\mathcal{E}}_i$. A zero-length chain is allowed. If ${\mathcal{E}}_i$ is empty then $X_i$ and the relation are empty. We call this the $E$ overlap chain relation. Throughout this definition, “the graph in def-e-graph-and-co-e-graph” means the $E$-graph defined there, not the co-$E$ graph.

The pattern graph of a pure blockade supplies the following definition: Let $\mathcal{B}=(B_1,\dots ,B_t)$ be a pure blockade in a graph $G$. Its pattern graph is the graph $P(\mathcal{B})$ with vertex set $[t]$ in which $i$ and $j$ are adjacent exactly when $B_i$ is complete to $B_j$. Because the blockade is pure, every unordered pair of distinct blocks is either complete or anticomplete, so this graph is well defined. A pattern graph is called $P_4$-free when it contains no induced four-vertex path.

The $E$-graph and co-$E$ supplies the following definition: The $E$-graph is the graph on vertices $\{p_1,p_2,p_3,p_4,p_5,q\}$ with edge set $\{p_1{p}_2,p_2{p}_3,p_3{p}_4,p_4{p}_5,p_{3}q\}\mathrm{.}$ Thus $p_1{p}_2{p}_3{p}_4{p}_5$ is a five-vertex path and $q$ is a leaf attached to its middle vertex $p_3$. The co-$E$ graph is the complement of this graph.

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 $X_i\mathrm{\ne }\mathrm{\varnothing }$.