# HG changeset patch
# User Madarasi Peter
# Date 1480484668 -3600
# Node ID 80d56dee41d9fd32fc747579a8c78d7fb9bd88e3
# Parent  b9a8744c5efcc545dc70e2e27f31f3db8b4e35f3
nth

diff -r b9a8744c5efc -r 80d56dee41d9 damecco.tex
--- a/damecco.tex	Wed Nov 30 06:21:19 2016 +0100
+++ b/damecco.tex	Wed Nov 30 06:44:28 2016 +0100
@@ -296,7 +296,7 @@
   V_{1} \longrightarrow V_{2}$ bijection, for which the
   following is true:
 \begin{center}
-$\forall u\in{V_{1}} : \mathcal{L}(u)=\mathcal{L}(\mathfrak{m}(u))$ and
+$\forall u\in{V_{1}} : \mathcal{L}(u)=\mathcal{L}(\mathfrak{m}(u))$ and\\
 $\forall u,v\in{V_{1}} : (u,v)\in{E_{1}} \Leftrightarrow (\mathfrak{m}(u),\mathfrak{m}(v))\in{E_{2}}$
 \end{center}
 \end{definition}
@@ -306,8 +306,7 @@
   V_{1}\longrightarrow V_{2}$ injection, for which the
   following is true:
 \begin{center}
-$\forall u\in{V_{1}} : \mathcal{L}(u)=\mathcal{L}(\mathfrak{m}(u))$ and
-$\mathbb{M}$
+$\forall u\in{V_{1}} : \mathcal{L}(u)=\mathcal{L}(\mathfrak{m}(u))$ and\\
 $\forall u,v \in{V_{1}} : (u,v)\in{E_{1}} \Rightarrow (\mathfrak{m}(u),\mathfrak{m}(v))\in E_{2}$
 \end{center}
 \end{definition}
@@ -380,7 +379,7 @@
 \end{definition}
 
 \begin{definition}
-Let \textbf{extend}$(\mathfrak{m},(u,v))$ denote the function $f : \mathfrak{D}(\mathfrak{m})\cup\{u\}\longrightarrow\mathfrak{R}(\mathfrak{m})\cup\{v\}$, for which $\forall w\in \mathfrak{D}(\mathfrak{m}) : \mathfrak{m}(w)=f(w)$ and $f(u)=v$ holds. Where $u\notin\mathfrak{D}(\mathfrak{m})$ and $v\notin\mathfrak{R}(\mathfrak{m})$, otherwise $extend(\mathfrak{m},(u,v))$ is undefined.
+Let \textbf{extend}$(\mathfrak{m},(u,v))$ denote the function $f : \mathfrak{D}(\mathfrak{m})\cup\{u\}\longrightarrow\mathfrak{R}(\mathfrak{m})\cup\{v\}$, for which $\forall w\in \mathfrak{D}(\mathfrak{m}) : \mathfrak{m}(w)=f(w)$ and $f(u)=v$ holds. Where $u\in V_1\setminus\mathfrak{D}(\mathfrak{m})$ and $v\in V_2\setminus\mathfrak{R}(\mathfrak{m})$, otherwise $extend(\mathfrak{m},(u,v))$ is undefined.
 \end{definition}
 
 \begin{notation}
@@ -816,10 +815,10 @@
 Being aware of Claim~\ref{claim:claimCoverFromLeft}, the
 task is not to maintain the candidate set, but to generate the
 candidate nodes in $G_{2}$ for a given node $u\in V_{1}$.  In
-case of any of the three problem types and a mapping $M$ if a node $v\in
+case of any of the three problem types and a mapping $\mathfrak{m}$, if a node $v\in
 V_{2}$ is a potential pair of $u\in V_{1}$, then $\forall
-u'\in V_{1} : (u,u')\in
-E_{1}\ and\ u'\ is\ covered\ by\ M\ \Rightarrow (v,Pair(M,u'))\in
+u'\in \mathfrak{D}(\mathfrak{m}) : (u,u')\in
+E_{1}\Rightarrow (v,\mathfrak{m}(u'))\in
 E_{2}$. That is, each covered neighbour of $u$ has to be mapped to
 a covered neighbour of $v$.