Robust expanders
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The following definitions follow closely those in [@kuhn2014robust]. Throughout, assume $G$ is an $n$-vertex graph. [**Robust expanders and bipartite robust expanders**]{} - Given an $n$-vertex graph $G$, and $S \subseteq V(G)$ and parameters $0<\nu\leq \tau<1$, we define the *$\nu$-robust neighbourhood* of $S$ to be ${\text{RN}}_{\nu,G}(S):=\{v\in G\mid d_S(v)\geq \nu n\}$. We say $G$ is a *robust $(\nu,\tau)$-expander* if for all $S\subseteq V(G)$ with $\tau n\leq |S| \leq (1-\tau)n$ we have $|{\text{RN}}_{\nu, G}(S)|\geq |S|+\nu n$. We say $G$ is a *bipartite robust $(\nu,\tau)$-expander* with bipartition $A,B$ if $A,B$ is a partition of $V(G)$ and for every $S\subseteq A$ with $\tau|A|\leq |S|\leq (1-\tau)|A|$ we have $|{\text{RN}}_{\nu,G}(S)|\geq|S|+\nu n$. Note that the order of $A$ and $B$ matters here.

[**Robust expander components and bipartite robust expander components**]{} - Given $0<\rho<1$ and an $n$-vertex graph $G$, we say that $U\subseteq V(G)$ is a *$\rho$-component* if $|U|\geq \sqrt{\rho}n$ and $e_G(U,\overline{U})\leq \rho n^2$, where as usual $\overline{U}:=V(G)\setminus U$. We say that $U$ is *$\rho$-close to bipartite* with bipartition $A,B$ if $A,B$ is a partition of $U$, $|A|,|B|\geq \sqrt{\rho}n$, $||A|-|B||\leq \rho n$, and $e_G(A,\overline{B})+e_G(B,\overline{A})\leq \rho n^2$. We will sometimes call a graph a $\rho$-component or $\rho$-close to bipartite if this holds for its vertex set. We 