Purity propagates through E overlap chains

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.

Edges between disjoint vertex sets; complete, anticomplete, pure and mixed pairs supplies the following definition: Let $G$ be a finite simple graph and let $A,B\subseteq V(G)$ be disjoint. An edge between $A$ and $B$ is an edge $ab$ with $a\in A$ and $b\in B$. The pair $(A,B)$ is: - complete when every $a\in A$ is adjacent to every $b\in B$; - anticomplete when no $a\in A$ is adjacent to any $b\in B$; - pure when it is complete or anticomplete; and - mixed when it is neither complete nor anticomplete. Adjacency is the symmetric edge relation of $G$ (def-finite-simple-graph, def-graph-adjacency-incidence-neighbourhood-and-degree). If $A=\mathrm{\varnothing }$ or $B=\mathrm{\varnothing }$, the pair is both complete and anticomplete, hence pure and not mixed.