$\varrho(G)$ of~$G$ as the average number of edges per vertex, i.e., $m(G)/n(G)$. The
edge density $\varrho(\cal C)$ of the class is then defined as the infimum of $\varrho(G)$ over all $G\in\cal C$.
-\thmn{Mader \cite{mader:dens}}\id{maderthm}%
+\obs
+Let us consider a~non-trivial minor-closed class~${\cal C} = \Forb({\cal H})$
+and a~graph $X\in{\cal H}$ with the minimum number of vertices.
+Obviously, $\Forb({\cal H}) \subseteq \Forb(X)$, because excluding additional
+minors cannot make the class richer. Also, if we denote the number of vertices
+of~$X$ by~$k$, we have $X\minorof K_k$ and hence $\Forb(X) \subseteq \Forb(K_k)$.
+When we put these two inclusions together, we get ${\cal C} \subseteq \Forb(K_k)$ and
+so $\varrho({\cal C}) \le \varrho(\Forb(K_k))$. It is therefore sufficient to
+bound the density of classes that exclude a~single complete graph.
+Moreover, our parameter~$k$ is equal to the well-known Hadwiger number:
+
+\defn\id{hadwiger}%
+The \df{Hadwiger number} $H(G)$ is the smallest~$k$ such that the complete
+graph~$K_k$ is not a~minor of~$G$. We can easily extend it to graph classes:
+$H({\cal C})$ is the minimum of~$H(G)$ over all~$G\in{\cal C}$.
+
+\cor
+$\varrho({\cal C}) \le \varrho(\Forb(K_{H({\cal C})}))$
+for any non-trivial minor-closed class~${\cal C}$.
+
+\thmn{Mader \cite{mader:dens}, see also Lemma 3.5.1 in Diestel \cite{diestel:gt}}\id{maderthm}%
For every $k\in{\bb N}$ there exists $h(k)\in{\bb R}$ such that every graph
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 of average
-degree at least~$2^m$ contains a~subdivision of some graph with $k$~vertices
-and $m$~edges (for $k\le m\le {k\choose 2}$). When we reach $m={k\choose 2}$, the theorem follows
-as the only graph with~$k$ vertices and~$k\choose 2$ edges is~$K_k$.
-
-The base case $m=k$: Let us observe that when the average degree
-is~$a$, removing any vertex of degree less than~$a/2$ does not decrease the
-average degree. A~graph with $a\ge 2^k$ therefore has a~subgraph
-with minimum degree $\delta\ge a/2=2^{k-1}$. Such subgraph contains
-a~cycle on more than~$\delta$ vertices, in other words a~subdivision of
-the cycle~$C_k$.
-
-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 the subgraph $G[U]$
-induced by~$U$ is connected and the graph $G\sgc U$ ($G$~with $U$~contracted to
-a~single vertex) has average degree at least~$2^m$ (such~$U$ exists, because
-$G=G\sgc U$ 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 $k$~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.
+A~non-trivial minor-closed class ${\cal C}$ has density $\varrho({\cal C}) \le 2h(k)$,
+where~$h$ is the function from the previous theorem and $k=H({\cal C})$ is the
+Hadwiger number of the class.
\proof
-Let~$\cal C$ be any such class, $X$~its excluded minor with the smallest number
-of vertices~$x$.
-As $X\minorof K_x$, the class $\cal C$ is entirely contained 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 hence $K_x$ as a~minor.
+We already know that it is sufficient to prove the theorem for the case when
+${\cal C}$ excludes on the complete graph~$K_k$.
+
+We will prove the contrapositive. If $\varrho({\cal C}) > 2h(k)$, then there is some graph
+$G\in{\cal C}$ such that $\varrho(G) > 2h(k)$. This implies that the average degree
+of~$G$ is greater than~$h(k)$, so by the previous theorem $G$~contains a~subdivison
+of~$K_k$ and hence also~$K_k$ as a~minor.
\qed
+\para
+The Mader's original proof of Theorem \ref{maderthm} yields $h(k) \approx 2^{n^2}$, which is
+very coarse. It was however vastly improved later: Kostochka
+\cite{kostochka:lbh} and independently Thomason \cite{thomason:efc} have proven
+that an~average degree $\Omega(k\sqrt{\log k})$ is sufficient to enforce~$K_k$
+as a~minor and that this is the best what we can get. Their result implies:
+
+\cor
+$\varrho({\cal C}) = \O(k\sqrt{\log k})$ whenever ${\cal C}$ is a~minor-closed
+class of graphs and~$k=H({\cal C})$ is its Hadwiger number.
+
Let us return to the analysis of our algorithm.
\thmn{MST on minor-closed classes, Tarjan \cite{tarjan:dsna}}\id{mstmcc}%
-For any fixed non-trivial minor-closed class~$\cal C$ of graphs, the Contractive Bor\o{u}vka's
+For any 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.)
+$\O(n \cdot \varrho({\cal C}))$.
\proof
Following the proof for planar graphs (\ref{planarbor}), we denote the graph considered
Since each $G_i$ is produced from~$G_{i-1}$ by a sequence of edge contractions,
all $G_i$'s are minors of the input graph.\foot{Technically, these are multigraph contractions,
but followed by flattening, so they are equivalent to contractions on simple graphs.}
-So they also belong to~$\cal C$ and by the Density theorem $m_i\le \varrho({\cal C})\cdot n_i$.
-The time complexity is therefore $\sum_i \O(m_i) = \sum_i \O(n_i) = \O(\sum_i n/2^i) = \O(n)$.
+So they also belong to~$\cal C$ and thus $m_i\le \varrho n_i$, where $\varrho=\varrho({\cal C})$
+is the density of the class~${\cal C}$.
+The time complexity of the algorithm is therefore
+$\sum_i \O(m_i) = \sum_i \O(\varrho n_i) = \O(\varrho\cdot\sum_i n/2^i) = \O(n\varrho)$.
\qed
\paran{Local contractions}\id{nobatch}%
\endalgo
\thm
-When $\cal C$ is a minor-closed class of graphs with density~$\varrho$, the
+When $\cal C$ is a~minor-closed class of graphs with density~$\varrho$, the
Local Bor\o{u}vka's Algorithm with the parameter~$t$ set to~$4\varrho$
-finds the MST of any graph from this class in time $\O(n)$. (The constant
-in the~$\O$ depends on~the class.)
+finds the MST of any graph from this class in time $\O(n\varrho)$.
\proof
Let us denote by $G_i$, $n_i$ and $m_i$ the graph considered by the
$m_i\le \varrho n_i \le \varrho n/2^i$.
We will show that the $i$-th iteration is carried out in time $\O(m_i)$.
Steps 5 and~6 run in time $\O(\deg(v))=\O(t)$ for each~$v$, so summed
-over all $v$'s they take $\O(tn_i)$, which is $\O(n_i)$ for a fixed class~$\cal C$.
+over all $v$'s they take $\O(tn_i)$.
Flattening takes $\O(m_i)$ as already noted in the analysis of the Contracting
Bor\o{u}vka's Algorithm (see \ref{contiter}).
-The whole algorithm therefore runs in time $\O(\sum_i m_i) = \O(\sum_i n/2^i) = \O(n)$.
+The whole algorithm therefore runs in time $\O(\sum_i m_i) = \O(\varrho\cdot\sum_i n/2^i) = \O(n\varrho)$.
\qed
\paran{Back to planar graphs}%
\figure{hexangle.eps}{\epsfxsize}{The construction from Remark~\ref{hexa}}
-\rem
-The bound on the average degree needed to enforce a~$K_k$ minor, which we get from Theorem \ref{maderthm},
-is very coarse. Kostochka \cite{kostochka:lbh} and independently Thomason \cite{thomason:efc}
-have proven that an~average degree $\Omega(k\sqrt{\log k})$ is sufficient and that this
-is the best what we can get.
-
\rem
Minor-closed classes share many other interesting properties, for example bounded chromatic
numbers of various kinds, as shown by Theorem 6.1 of \cite{nesetril:minors}. We can expect
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