The $E$-graph and Bird singleton families are leaf-reducible

In $E$ the only edge incident with $q$ is $p_{3}q$, by [L1]. Hence $q$ has degree one and is a leaf, and $E\setminus \{q\}$ is the induced subgraph on $\{p_1,p_2,p_3,p_4,p_5\}$ with exactly the four edges $p_1{p}_2,p_2{p}_3,p_3{p}_4,p_4{p}_5$.

In Bird the only edge incident with $w$ is $yw$, by [L2]. Hence $w$ has degree one and is a leaf, and $\mathrm{B}\mathrm{i}\mathrm{r}\mathrm{d}\setminus \{w\}$ is the induced subgraph on $\{x_1,x_2,x_3,y,z\}$ with edge set $\{x_1{x}_2,x_2{x}_3,x_1{x}_3,x_{1}y,x_{2}z\}$, which is exactly the bull of [L3].

Take $\mathcal{F}=\{\mathrm{B}\mathrm{i}\mathrm{r}\mathrm{d}\}$ and the member $\mathrm{B}\mathrm{i}\mathrm{r}\mathrm{d}$ with the leaf $w$ of step 1.2. Then $\mathrm{B}\mathrm{i}\mathrm{r}\mathrm{d}\setminus \{w\}$ is the bull by step 1.2, so the modified family of [L5] is the singleton $\{\text{bull}\}$, whose Erdős-Hajnal property is [L10] read through the family terminology of [L11]; both phrases describe the same class of bull-free graphs. Hence $\{\mathrm{B}\mathrm{i}\mathrm{r}\mathrm{d}\}$ is leaf-reducible.

The map $\phi (i):=p_{i+1}$ for $i=0,1,2,3,4$ is a bijection from $V(P_5)$ onto $\{p_1,\dots ,p_5\}$ whose four edges $\{i,i+1\}$ of [L4] correspond to the four edges $p_{i+1}{p}_{i+2}$ listed in step 1.1, and no other pairs are edges on either side. A bijection matching adjacency and nonadjacency is an isomorphism by [L7], so $E\setminus \{q\}\cong P_5$.

Consequently a finite graph $G$ is $(E\setminus \{q\})$-free if and only if it is $P_5$-free: composing an induced embedding of $P_5$ in $G$ with the inverse of the isomorphism of step 2.2 yields an induced embedding of $E\setminus \{q\}$ in $G$, and composing an induced embedding of $E\setminus \{q\}$ with that isomorphism yields an induced embedding of $P_5$.

By [L9] the class of $P_5$-free graphs has an Erdős-Hajnal constant; step 3.1 identifies it with the class of $(E\setminus \{q\})$-free graphs, so that class also has a constant, and [L11] makes the singleton family $\{E\setminus \{q\}\}$ a family with the Erdős-Hajnal property.

Take $\mathcal{F}=\{E\}$ and the member $H=E$ with the leaf $q$ of step 1.1. Then $H\setminus \{q\}=E\setminus \{q\}$, so the modified family of [L5] is $\{E\setminus \{q\}\}\cup (\{E\}\setminus \{E\})=\{E\setminus \{q\}\}$, which has the Erdős-Hajnal property by step 4.1. Hence $\{E\}$ is leaf-reducible.

The two singleton families are leaf-reducible, the deleted graph is $P_5$ in the $E$ case and the bull in the Bird case, and the reduced singleton families $\{E\setminus \{q\}\}$ and $\{\mathrm{B}\mathrm{i}\mathrm{r}\mathrm{d}\setminus \{w\}\}$ have the Erdős-Hajnal property by steps 2.1 and 4.1. These are all the assertions of the statement.