1.1 --- a/damecco.tex	Wed Nov 30 06:21:19 2016 +0100
     1.2 +++ b/damecco.tex	Wed Nov 30 06:44:28 2016 +0100
     1.3 @@ -296,7 +296,7 @@
     1.4    V_{1} \longrightarrow V_{2}$ bijection, for which the
     1.5    following is true:
     1.6  \begin{center}
     1.7 -$\forall u\in{V_{1}} : \mathcal{L}(u)=\mathcal{L}(\mathfrak{m}(u))$ and
     1.8 +$\forall u\in{V_{1}} : \mathcal{L}(u)=\mathcal{L}(\mathfrak{m}(u))$ and\\
     1.9  $\forall u,v\in{V_{1}} : (u,v)\in{E_{1}} \Leftrightarrow (\mathfrak{m}(u),\mathfrak{m}(v))\in{E_{2}}$
    1.10  \end{center}
    1.11  \end{definition}
    1.12 @@ -306,8 +306,7 @@
    1.13    V_{1}\longrightarrow V_{2}$ injection, for which the
    1.14    following is true:
    1.15  \begin{center}
    1.16 -$\forall u\in{V_{1}} : \mathcal{L}(u)=\mathcal{L}(\mathfrak{m}(u))$ and
    1.17 -$\mathbb{M}$
    1.18 +$\forall u\in{V_{1}} : \mathcal{L}(u)=\mathcal{L}(\mathfrak{m}(u))$ and\\
    1.19  $\forall u,v \in{V_{1}} : (u,v)\in{E_{1}} \Rightarrow (\mathfrak{m}(u),\mathfrak{m}(v))\in E_{2}$
    1.20  \end{center}
    1.21  \end{definition}
    1.22 @@ -380,7 +379,7 @@
    1.23  \end{definition}
    1.24  
    1.25  \begin{definition}
    1.26 -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.
    1.27 +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.
    1.28  \end{definition}
    1.29  
    1.30  \begin{notation}
    1.31 @@ -816,10 +815,10 @@
    1.32  Being aware of Claim~\ref{claim:claimCoverFromLeft}, the
    1.33  task is not to maintain the candidate set, but to generate the
    1.34  candidate nodes in $G_{2}$ for a given node $u\in V_{1}$.  In
    1.35 -case of any of the three problem types and a mapping $M$ if a node $v\in
    1.36 +case of any of the three problem types and a mapping $\mathfrak{m}$, if a node $v\in
    1.37  V_{2}$ is a potential pair of $u\in V_{1}$, then $\forall
    1.38 -u'\in V_{1} : (u,u')\in
    1.39 -E_{1}\ and\ u'\ is\ covered\ by\ M\ \Rightarrow (v,Pair(M,u'))\in
    1.40 +u'\in \mathfrak{D}(\mathfrak{m}) : (u,u')\in
    1.41 +E_{1}\Rightarrow (v,\mathfrak{m}(u'))\in
    1.42  E_{2}$. That is, each covered neighbour of $u$ has to be mapped to
    1.43  a covered neighbour of $v$.
    1.44