Complete nonedge pairs force purity on induced E graphs

A mixed vertex on E is pure on both terminal edges 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 $p_1{p}_2,p_2{p}_3,p_3{p}_4,p_4{p}_5,p_{3}q$. If $u$ is mixed on $S$, it is nevertheless pure to both terminal edges $\{p_1,p_2\}$ and $\{p_4,p_5\}$.

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.

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.

Combs in a graph supplies the following definition: Let $\mathrm{\ell }\in \mathbb{N}$ with $\mathrm{\ell }\ge 1$, and let $w>0$. An $(\mathrm{\ell },w)$-comb in a graph $G$ is a sequence of pairs $((a_i,B_i):i\in [\mathrm{\ell }])$ satisfying the conditions below. Here a vertex $a$ is complete to (respectively, anticomplete to) a set $B$ when the pair $(\{a\},B)$ is complete (respectively, anticomplete) in the sense of def-edges-between-sets-and-pure-mixed-pairs. 1. $(B_1,\dots ,B_{\mathrm{\ell }})$ is an $(\mathrm{\ell },w)$-blockade; 2. the vertices $a_1,\dots ,a_{\mathrm{\ell }}$ are distinct; 3. the set $\{a_1,\dots ,a_{\mathrm{\ell }}\}$ is disjoint from every block $B_i$; and 4. for every $i\in [\mathrm{\ell }]$, the vertex $a_i$ is complete to $B_i$; and 5. for all distinct $i,j\in [\mathrm{\ell }]$, the vertex $a_i$ is anticomplete to $B_j$. The vertices $a_i$ are the teeth of the comb.