Saving text which will be deleted :)

diff --git a/mst.tex b/mst.tex
index 354e0d60e73a57ff7fbbfdee3e2c466f7f0346fe..7642ddb3e58519683ff4c9d8eea647dcc2303bc5 100644 (file)
--- a/mst.tex
+++ b/mst.tex
@@ -459,10 +459,10 @@ use the multigraph version and show that we can easily reduce the multigraph
 to a simple graph later. (See \thmref{contract} for the exact definitions.)
 
 We only need to be able to map edges of the contracted graph to the original
-edges, so each edge will carry a unique label $l(e)$ which will be preserved by
+edges, so each edge will carry a unique label $\ell(e)$ which will be preserved by
 contractions.
 
-\lemman{Flattening a multigraph}%
+\lemman{Flattening a multigraph}\thmid{flattening}%
 Let $G$ be a multigraph and $G'$ its subgraph such that all loops have been
 removed and each bundle of parallel edges replaced by its lightest edge.
 Then $G'$~has the same MST as~$G$.
@@ -484,10 +484,10 @@ modifications.
 \algo
 \algin A~graph~$G$ with an edge comparison oracle.
 \:$T\=\emptyset$.
-\:$l(e)\=e$ for all edges~$e$. \cmt{Initialize the labels.}
+\:$\ell(e)\=e$ for all edges~$e$. \cmt{Initialize the labels.}
 \:While $n(G)>1$:
 \::For each vertex $v_i$ of~$G$, let $e_i$ be the lightest edge incident to~$v_i$.
-\::$T\=T\cup \{ l(e_i) \}$. \cmt{Remember labels of all selected edges.}
+\::$T\=T\cup \{ \ell(e_i) \}$. \cmt{Remember labels of all selected edges.}
 \::Contract $G$ along all edges $e_i$, inheriting labels and weights.
 \::Flatten $G$, removing parallel edges and loops.
 \algout Minimum spanning tree~$T$.
@@ -565,13 +565,33 @@ Graph contractions are a very powerful tool and they can be used in other MST
 algorithms as well. The following lemma shows the gist:
 
 \lemman{Contraction of MST edges}
-Let $G$ be a weighted multigraph, $e$~an arbitrary edge of~$\mst(G)$, $G/e$ the multigraph
-produced by contracting $G$ along~$e$ and $\pi$ the bijection between edges of~$G-e$, and
+Let $G$ be a weighted graph, $e$~an arbitrary edge of~$\mst(G)$, $G/e$ the multigraph
+produced by contracting $G$ along~$e$, and $\pi$ the bijection between edges of~$G-e$ and
 their counterparts in~$G/e$. Then: $$\mst(G) = \pi^{-1}[\mst(G/e)] + e.$$
 
 \proof
+% We seem not to need this lemma for multigraphs...
+%If there are any loops or parallel edges in~$G$, we can flatten the graph. According to the
+%Flattening lemma (\thmref{flattening}), the MST stays the same and if we remove a parallel edge
+%or loop~$f$, then $\pi(f)$ would be removed when flattening~$G/e$, so $f$ never participates
+%in a MST.
+The right-hand side of the equality is a spanning tree of~$G$, let us denote it by~$T$ and
+the MST of $G/e$ by~$T'$. If $T$ were not minimum, there would exist a $T$-light edge~$f$ in~$G$.
+If the path $T[f]$ covered by~$f$ does not contain~$e$, then $\pi[T[f]]$ is a path covered by~$\pi(f)$
+in~$T'$. Otherwise $\pi(T[f]-e)$ is such a path. In both cases, $f$ is $T'$-light, which should
+contradict the minimality of~$T'$, but we need a multigraph version of Theorem \thmref{mstthm}
+which we did not prove.
+
+To avoid it, we consider the flattening~$F$ of~$G/e$ and apply the theorem to it.
+According to the Flattening lemma (\thmref{flattening}), $F$~has the same MST as~$G/e$
+and therefore it contains all edges of~$T'$. If $\pi(f)\not\in E(F)$, it
+has been removed in favor of some lighter edge~$f'$ and if $\pi(f)$ was
+$T$-light, then $f'$ is even so.
 \qed
 
+\rem
+In the previous algorithm, the role of the mapping~$\pi^{-1}$ is played by the edge labels~$\ell$.
+
 \section{Minor-closed graph classes}
 
 \section{Using Fibonacci heaps}