Many of the nice structural properties of planar graphs extend to
minor-closed classes, too (see \cite{diestel:gt} for a~nice overview
of this theory). The most important property is probably the characterization
-of such classes by forbidden minors.
+of such classes in terms of their forbidden minors.
\defn
For a~class~$\cal H$ of graphs we define $\Forb({\cal H})$ as the class
of graphs which do not contain any of the graphs in~$\cal H$ as a~minor.
We will call $\cal H$ the set of \df{forbidden (or excluded) minors} for this class.
+We will often abbreviate $\Forb(\{M_1,\ldots,M_n\})$ to $\Forb(M_1,\ldots,M_n)$.
\obs
For every~${\cal H}\ne\emptyset$, the class $\Forb({\cal H})$ is non-trivial
and closed on minors. This works in the opposite direction as well: for every
minor-closed class~$\cal C$ there is a~class $\cal H$ such that ${\cal C}=\Forb({\cal H})$.
-One such~$\cal H$ is the complement of~$\cal C$, but smaller sets can be found, too.
-For example, the planar graphs exclude exactly $K_5$ and~$K_{3,3}$ --- this follows
-from the Kuratowski's theorem (the theorem uses forbidden subdivisions, but while
-in general this is not the same as forbidden minors, it is for $K_5$ and $K_{3,3}$).
-The celebrated theorem by Robertson and Seymour guarantees that we can always find
-a~finite set of forbidden minors.
+One such~$\cal H$ is the complement of~$\cal C$, but smaller ones can be found, too.
+For example, the planar graphs can be equivalently described as the class $\Forb(K_5, K_{3,3})$
+--- this follows from the Kuratowski's theorem (the theorem speaks of forbidden
+subdivisions, but while in general this is not the same as forbidden minors, it
+is for $K_5$ and $K_{3,3}$). The celebrated theorem by Robertson and Seymour
+guarantees that we can always find a~finite set of forbidden minors.
\thmn{Excluded minors, Robertson \& Seymour \cite{rs:wagner}}
For every non-trivial minor-closed graph class~$\cal C$ there exists
a~finite set~$\cal H$ of graphs such that ${\cal C}=\Forb({\cal H})$.
\proof
-This theorem has been proven in the series of papers on graph minors
+This theorem has been proven in a~long series of papers on graph minors
culminating with~\cite{rs:wagner}. See this paper and follow the references
to the previous articles in the series.
\qed
\para
-We will make use of another important property --- the bounded density of minor-closed
-classes. The connection between minors and density dates back to Mader in the 1960's
-and it can be proven without use of the Robertson-Seymour theorem.
+For analysis of the contractive algorithm,
+we will make use of another important property --- the bounded density of
+minor-closed classes. The connection between minors and density dates back to
+Mader in the 1960's and it can be proven without use of the Robertson-Seymour
+theorem.
\defn\id{density}%
Let $\cal C$ be a class of graphs. We define its \df{edge density} $\varrho(\cal C)$
\thmn{Mader \cite{mader:dens}}
For every $k\in{\bb N}$ there exists $h(k)\in{\bb R}$ such that every graph
-with average degree at least~$h(k)$ contains a~subdivision of~$K_{k}$ as a~subgraph.
+of average degree at least~$h(k)$ contains a~subdivision of~$K_{k}$ as a~subgraph.
\proofsketch
(See Lemma 3.5.1 in \cite{diestel:gt} for a~complete proof in English.)
-Let us fix~$k$ and prove by induction on~$m$ that every graph with average
-degree $a\ge 2^m$ contains a~subdivision of some graph with $k$~vertices
+Let us fix~$k$ and prove by induction on~$m$ that every graph of average
+degree at least~$2^m$ contains a~subdivision of some graph with $k$~vertices
and ${k\choose 2}\ge m\ge k$~edges. For $m={k\choose 2}$ the theorem follows
as the only graph with~$k$ vertices and~$k\choose 2$ edges is~$K_k$.
Induction step: Let~$G$ be a~graph with average degree at least~$2^m$ and
assume that the theorem already holds for $m-1$. Without loss of generality,
-$G$~is connected. Consider a~maximal set $U\subseteq V$ such that $G[U]$ is connected
-and the graph $G.U$ ($G$~with $U$~contracted to a~single vertex) has average
-degree at least~$2^m$ (such~$U$ exists, because $G=G.U$ whenever $\vert U\vert=1$).
-Now consider the subgraph~$H$ induced in~$G$ by the
-neighbors of~$U$. Every $v\in V(H)$ must have degree at least~$2^{m-1}$
-(otherwise we can add this vertex to~$U$, contradicting its maximality), so by
-the induction hypothesis $H$ contains a~subdivision of some graph~$R$ with
-$r$~vertices and $m-1$ edges. Any two non-adjacent vertices of~$R$ can be
-connected in the subdivision by a~path lying entirely in~$G[U]$, which reveals
-a~subdivision of a~graph on $m$~vertices. \qed
+$G$~is connected. Consider a~maximal set $U\subseteq V$ such that the subgraph $G[U]$
+induced by~$U$ is connected and the graph $G.U$ ($G$~with $U$~contracted to
+a~single vertex) has average degree at least~$2^m$ (such~$U$ exists, because
+$G=G.U$ whenever $\vert U\vert=1$). Now consider the subgraph~$H$ induced
+in~$G$ by the neighbors of~$U$. Every $v\in V(H)$ must have $\deg_H(v) \ge 2^{m-1}$,
+as otherwise we can add this vertex to~$U$, contradicting its
+maximality. By the induction hypothesis, $H$ contains a~subdivision of some
+graph~$R$ with $r$~vertices and $m-1$ edges. Any two non-adjacent vertices
+of~$R$ can be connected in the subdivision by a~path lying entirely in~$G[U]$,
+which reveals a~subdivision of a~graph with $m$~edges. \qed
\thmn{Density of minor-closed classes, Mader~\cite{mader:dens}}
Every non-trivial minor-closed class of graphs has finite edge density.
\proof
Let~$\cal C$ be any such class, $X$~its smallest excluded minor and $x=n(X)$.
-As $H\minorof K_x$, the class $\cal C$ entirely lies in ${\cal C}'=\Forb(\{K_x\})$, so
+As $H\minorof K_x$, the class $\cal C$ entirely lies in ${\cal C}'=\Forb(K_x)$, so
$\varrho({\cal C}) \le \varrho({\cal C}')$ and therefore it suffices to prove the
theorem for classes excluding a~single complete graph~$K_x$.
We will show that $\varrho({\cal C})\le 2h(x)$, where $h$~is the function
from the previous theorem. If any $G\in{\cal C}$ had more than $2h(x)\cdot n(G)$
edges, its average degree would be at least~$h(x)$, so by the previous theorem
-$G$~would contain a~subdivision of~$K_x$ and therefore $K_x$ as a~minor.
+$G$~would contain a~subdivision of~$K_x$ and hence $K_x$ as a~minor.
\qed
\rem
-See also Theorem 6.1 in \cite{nesetril:minors}, which also lists some other equivalent conditions.
+Minor-closed classes share many other interesting properties, as shown for
+example by Theorem 6.1 of \cite{nesetril:minors}.
\thmn{MST on minor-closed classes \cite{mm:mst}}\id{mstmcc}%
-For any fixed non-trivial minor-closed class~$\cal C$ of graphs, Algorithm \ref{contbor} finds
-the MST of any graph in this class in time $\O(n)$. (The constant hidden in the~$\O$
-depends on the class.)
+For any fixed non-trivial minor-closed class~$\cal C$ of graphs, the Contractive Bor\o{u}vka's
+algorithm (\ref{contbor}) finds the MST of any graph of this class in time
+$\O(n)$. (The constant hidden in the~$\O$ depends on the class.)
\proof
Following the proof for planar graphs (\ref{planarbor}), we denote the graph considered
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