All E neighbourhood patterns under a complete nonedge pair

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 $c$.

The path-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 $H=\{a,b,c,d\}$ induces the path $a-b-c$ and isolated vertex $d$, then $u$ cannot be adjacent to $d$ and two consecutive vertices of the path and nonadjacent to the remaining endpoint.

Complete nonedge pairs force purity on induced E graphs 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. Let $S=\{p_1,p_2,p_3,p_4,p_5,q\}$ induce $E$, with edges exactly $p_1{p}_2,p_2{p}_3,p_3{p}_4,p_4{p}_5,p_{3}q$. Then $u$ is pure to $S$. In particular, in an $(\mathrm{\ell },w)$-comb with an outside vertex $v$ complete to all blocks and anticomplete to all teeth, each $u\in B_k$, $k\mathrm{\ne }i$, is pure to every induced $E$ in $B_i$.

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.