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\section{Notation}
I'm largely following Robson's notation \cite{robson}. For a graph $G=(V,E)$ and vertex $v\in V$, and edge $e\in E$, we write $d(v)$ for the degree of the vertex. $N(v)$ is the set of $v$'s neighbours, and $\iN(v)=N(v)\cup\{v\}$, $v$'s `inclusive neighbourhood'. Then $N^2(v)$ is the set of second order neighbours of $v$, excluding $v$: $N^2(v)=\{v_2 \in N(v_1) \mid v_1\in N(v)\} \setminus \{v\}$. If $\iN(v)\subseteq\iN(w)$, $v$ is said to dominate $w$, notation: $v>w$.
We write `subgraph' for `vertex-induced subgraph', and $G\ex U$ for the subgraph induced on $G$ by $U$. For $v\in V$, we write $G\ex v$ for $G\ex\{v\}$. We use $e(v,w)$ for the predicate $(v,w)\in E$. We're strictly talking about non-directed graphs, so this is equivalent with $(w,v)\in E$.
Finally, We write `m.i.s.' for `maximum independent set' and $\ms(\cdots)$ for its size, the graph's independence number. The size of a graph or an m.i.s. is defined as the number of vertices and is written as $|G|$.
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