E overlap classes form an anticonnected partition

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.

Anticonnected graphs and anticonnected components supplies the following definition: A graph $G$ is anticonnected, or co-connected, when its complement $\bar{G}$ is connected (def-connected-graph-and-connected-component, def-graph-isomorphism-and-complement). An anticonnected component, or anticomponent, of $G$ is a vertex set $A\subseteq V(G)$ that is the vertex set of a connected component of $\bar{G}$. Equivalently, $G[A]$ is anticonnected and $A$ is inclusion-maximal with that property (def-subgraph-induced-subgraph-and-spanning-subgraph). Under the library convention, the null graph is not anticonnected, while a one-vertex graph is anticonnected.

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.