embeddable in any fixed surface. Many nice properties of planar graphs extend
to these classes, too, most notable the linearity of the number of edges.
-\defn
+\defn\thmid{density}%
Let $\cal C$ be a class of graphs. We define its \df{edge density} $\varrho(\cal C)$
to be the infimum of all~$\varrho$'s such that $\vert E(G) \vert \le \varrho\cdot\vert V(G)\vert$
holds for every $G\in\cal C$.
\n{$G.e$}{simple graph contraction \[simpcont]}
\n{$\alpha(n)$}{the inverse Ackermann's function}
\n{$f[X]$}{function applied to a set: $f[X]:=\{ f(x) ; x\in X \}$.}
+\n{$\varrho({\cal C})$}{edge density of a graph class~$\cal C$ \[density]}
}
\section{Multigraphs and contractions}