A separated anticonnected block pair forbids mixing in one direction

The edge-plus-isolate co-Bird obstruction supplies the following statement: Let $G$ be a finite simple co-Bird-free graph. Let $x,y,u$ be distinct vertices outside the indicated induced subgraph, with $xy\mathrm{\notin }E(G)$, $ux\in E(G)$ and $uy\mathrm{\notin }E(G)$, and with $x,y$ complete to that subgraph. If $H=\{a,b,c\}$ induces just the edge $ab$, then $u$ cannot be mixed on $\{a,b\}$ and nonadjacent to $c$.

A vertex mixed on an anticonnected set yields opposite adjacency on a nonedge supplies the following statement: Let $G$ be a finite graph, let $A\subseteq V(G)$ be anticonnected, and let $v\in V(G)\setminus A$ be mixed on $A$. Then there exist distinct vertices $b,b^{\mathrm{\prime }}\in A$ such that $b{b}^{\mathrm{\prime }}\mathrm{\notin }E(G),vb\in E(G),v{b}^{\mathrm{\prime }}\mathrm{\notin }E(G)\mathrm{.}$

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.