More wordsmithing.

diff --git a/mst.tex b/mst.tex
index 8d3f18a21e6f09daef274a8999a111dcaf85142e..445b6ce069cf1ca5455d6148ea8a24898a402fe4 100644 (file)
--- a/mst.tex
+++ b/mst.tex
@@ -168,7 +168,7 @@ minimum spanning trees according to the Cayley's formula \cite{cayley:trees}).
 However, as the following theorem shows, this is possible only if the weight
 function is not injective.
 
-\thmn{MST uniqueness}%
+\thmn{Uniqueness of MST}%
 If all edge weights are distinct, then the minimum spanning tree is unique.
 
 \proof
@@ -181,6 +181,10 @@ we know that $w(T_1)=w(T_2)$, so the exchange sequence must be empty and indeed
 $T_1$ and $T_2$ must be identical.
 \qed
 
+\nota\id{mstnota}%
+When $G$ is a graph with distinct edge weights, we will use $\mst(G)$ to denote
+its unique minimum spanning tree.
+
 \rem\id{edgeoracle}%
 To simplify the description of MST algorithms, we will expect that the weights
 of all edges are distinct and that instead of numeric weights (usually accompanied
@@ -192,13 +196,8 @@ minimum spanning trees, the unique MST of the new graph will still be a MST of t
 original graph. In the few cases where we need a more concrete input, we will
 explicitly state so.
 
-\nota\id{mstnota}%
-When $G$ is a graph with distinct edge weights, we will use $\mst(G)$ to denote
-its unique minimum spanning tree.
-
 Another useful consequence is that whenever two graphs are isomorphic and the
-isomorphism preserves weight order, the isomorphism applies to their MST's
-as well:
+isomorphism preserves the relative order of weights, the isomorphism applies to their MST's as well:
 
 \defn
 A~\df{monotone isomorphism} of two weighted graphs $G_1=(V_1,E_1,w_1)$ and
@@ -207,7 +206,7 @@ for each $u,v\in V_1: uv\in E_1 \Leftrightarrow \pi(u)\pi(v)\in E_2$ and
 for each $e,f\in E_1: w_1(e)<w_1(f) \Leftrightarrow w_2(\pi[e]) < w_2(\pi[f])$.
 
 \lemman{MST of isomorphic graphs}\id{mstiso}%
-Let~$G_1$ and $G_2$ be two weighted graphs with unique edge weights and $\pi$
+Let~$G_1$ and $G_2$ be two weighted graphs with distinct edge weights and $\pi$
 their monotone isomorphism. Then $\mst(G_2) = \pi[\mst(G_1)]$.
 
 \proof