1.1 --- a/damecco.tex	Wed Nov 23 21:45:11 2016 +0100
     1.2 +++ b/damecco.tex	Thu Nov 24 22:09:36 2016 +0100
     1.3 @@ -402,7 +402,7 @@
     1.4  \end{definition}
     1.5  
     1.6  \begin{definition}
     1.7 -Mapping M \textbf{covers} a node v, if there exists a pair in M, which
     1.8 +Mapping $M$ \textbf{covers} a node v, if there exists a pair in M, which
     1.9  contains v.
    1.10  \end{definition}
    1.11  
    1.12 @@ -435,42 +435,14 @@
    1.13    (Pair(M,u),Pair(M,v))\in E_{large}$.
    1.14  \end{center}
    1.15  
    1.16 -\begin{definition}
    1.17 -A set of whole mappings is called \textbf{problem type}.
    1.18 -\end{definition}
    1.19  Throughout the paper, $\mathbf{PT}$ denotes a generic problem type
    1.20 -which can be substituted by any problem type.
    1.21 -
    1.22 -A whole mapping $W\mathbf{\ is\ of\ type\ PT}$, if $W\in PT$. Using
    1.23 -this notations, VF2 searches a whole mapping $W$ of type $PT$.
    1.24 -
    1.25 -For example the problem type of graph isomorphism problem is the
    1.26 -following.  A whole mapping $W$ is in $\mathbf{ISO}$, iff the
    1.27 -bijection represented by $W$ satisfies Definition~\ref{sec:ismorphic}.
    1.28 -The subgraph- and induced subgraph matching problems can be formalized
    1.29 -in a similar way. Let their problem types be denoted as $\mathbf{SUB}$
    1.30 +which can be substituted by any of $\mathbf{ISO}$, $\mathbf{SUB}$
    1.31  and $\mathbf{IND}$.
    1.32  
    1.33  \begin{definition}
    1.34 -\label{expPT}
    1.35 -$PT$ is an \textbf{expanding problem type} if $\ \forall\ W\in
    1.36 -PT:\ \forall u_1,u_2\in V_{small}:\ (u_1,u_2)\in E_{small}\Rightarrow
    1.37 -(Pair(W,u_1),Pair(W,u_2))\in E_{large}$, that is each edge of
    1.38 -$G_{small}$ has to be mapped to an edge of $G_{large}$ for each
    1.39 -mapping in $PT$.
    1.40 -\end{definition}
    1.41 -
    1.42 -Note that $ISO$, $SUB$ and $IND$ are expanding problem types.
    1.43 -
    1.44 -This paper deals with the three problem types mentioned above only,
    1.45 -but the following generic definitions make it possible to handle other
    1.46 -types as well.  Although it may be challenging to find a proper
    1.47 -consistency function and an efficient cutting function.
    1.48 -
    1.49 -\begin{definition}
    1.50  Let M be a mapping. A logical function $\mathbf{Cons_{PT}}$ is a
    1.51  \textbf{consistency function by } $\mathbf{PT}$, if the following
    1.52 -holds. If there exists whole mapping $W$ of type PT for which
    1.53 +holds. If there exists whole mapping $W$ of type $PT$ for which
    1.54  $M\subseteq W$, then $Cons_{PT}(M)$ is true.
    1.55  \end{definition}
    1.56  
    1.57 @@ -478,7 +450,7 @@
    1.58  Let M be a mapping. A logical function $\mathbf{Cut_{PT}}$ is a
    1.59  \textbf{cutting function by } $\mathbf{PT}$, if the following
    1.60  holds. $\mathbf{Cut_{PT}(M)}$ is false if $M$ can be extended to a
    1.61 -whole mapping W of type PT.
    1.62 +whole mapping $W$ of type $PT$.
    1.63  \end{definition}
    1.64  
    1.65  \begin{definition}
    1.66 @@ -572,9 +544,7 @@
    1.67  
    1.68  \subsection{The candidate set P(s)}
    1.69  \label{candidateComputingVF2}
    1.70 -$P(s)$ is the set of the candidate pairs for inclusion in $M(s)$.
    1.71 -Suppose that $PT$ is an expanding problem type, see
    1.72 -Definition~\ref{expPT}.
    1.73 +Let $P(s)$ be the set of the candidate pairs for inclusion in $M(s)$.
    1.74  
    1.75  \begin{notation}
    1.76  Let $\mathbf{T_{small}(s)}:=\{u \in V_{small} : u$ is not covered by
    1.77 @@ -974,7 +944,7 @@
    1.78  Being aware of Claim~\ref{claim:claimCoverFromLeft}, the
    1.79  task is not to maintain the candidate set, but to generate the
    1.80  candidate nodes in $G_{large}$ for a given node $u\in V_{small}$.  In
    1.81 -case of an expanding problem type and $M$ mapping, if a node $v\in
    1.82 +case of any of the three problem type and a mapping $M$, if a node $v\in
    1.83  V_{large}$ is a potential pair of $u\in V_{small}$, then $\forall
    1.84  u'\in V_{small} : (u,u')\in
    1.85  E_{small}\ and\ u'\ is\ covered\ by\ M\ \Rightarrow (v,Pair(M,u'))\in