# HG changeset patch
# User Madarasi Peter
# Date 1480021776 -3600
# Node ID b45bac51110897e7d8593f596ccf88ae35f284f0
# Parent  a21760ed63d67771b2dd53891616db53110b15e7
Expanding PT part removed

diff -r a21760ed63d6 -r b45bac511108 damecco.tex
--- a/damecco.tex	Wed Nov 23 21:45:11 2016 +0100
+++ b/damecco.tex	Thu Nov 24 22:09:36 2016 +0100
@@ -402,7 +402,7 @@
 \end{definition}
 
 \begin{definition}
-Mapping M \textbf{covers} a node v, if there exists a pair in M, which
+Mapping $M$ \textbf{covers} a node v, if there exists a pair in M, which
 contains v.
 \end{definition}
 
@@ -435,42 +435,14 @@
   (Pair(M,u),Pair(M,v))\in E_{large}$.
 \end{center}
 
-\begin{definition}
-A set of whole mappings is called \textbf{problem type}.
-\end{definition}
 Throughout the paper, $\mathbf{PT}$ denotes a generic problem type
-which can be substituted by any problem type.
-
-A whole mapping $W\mathbf{\ is\ of\ type\ PT}$, if $W\in PT$. Using
-this notations, VF2 searches a whole mapping $W$ of type $PT$.
-
-For example the problem type of graph isomorphism problem is the
-following.  A whole mapping $W$ is in $\mathbf{ISO}$, iff the
-bijection represented by $W$ satisfies Definition~\ref{sec:ismorphic}.
-The subgraph- and induced subgraph matching problems can be formalized
-in a similar way. Let their problem types be denoted as $\mathbf{SUB}$
+which can be substituted by any of $\mathbf{ISO}$, $\mathbf{SUB}$
 and $\mathbf{IND}$.
 
 \begin{definition}
-\label{expPT}
-$PT$ is an \textbf{expanding problem type} if $\ \forall\ W\in
-PT:\ \forall u_1,u_2\in V_{small}:\ (u_1,u_2)\in E_{small}\Rightarrow
-(Pair(W,u_1),Pair(W,u_2))\in E_{large}$, that is each edge of
-$G_{small}$ has to be mapped to an edge of $G_{large}$ for each
-mapping in $PT$.
-\end{definition}
-
-Note that $ISO$, $SUB$ and $IND$ are expanding problem types.
-
-This paper deals with the three problem types mentioned above only,
-but the following generic definitions make it possible to handle other
-types as well.  Although it may be challenging to find a proper
-consistency function and an efficient cutting function.
-
-\begin{definition}
 Let M be a mapping. A logical function $\mathbf{Cons_{PT}}$ is a
 \textbf{consistency function by } $\mathbf{PT}$, if the following
-holds. If there exists whole mapping $W$ of type PT for which
+holds. If there exists whole mapping $W$ of type $PT$ for which
 $M\subseteq W$, then $Cons_{PT}(M)$ is true.
 \end{definition}
 
@@ -478,7 +450,7 @@
 Let M be a mapping. A logical function $\mathbf{Cut_{PT}}$ is a
 \textbf{cutting function by } $\mathbf{PT}$, if the following
 holds. $\mathbf{Cut_{PT}(M)}$ is false if $M$ can be extended to a
-whole mapping W of type PT.
+whole mapping $W$ of type $PT$.
 \end{definition}
 
 \begin{definition}
@@ -572,9 +544,7 @@
 
 \subsection{The candidate set P(s)}
 \label{candidateComputingVF2}
-$P(s)$ is the set of the candidate pairs for inclusion in $M(s)$.
-Suppose that $PT$ is an expanding problem type, see
-Definition~\ref{expPT}.
+Let $P(s)$ be the set of the candidate pairs for inclusion in $M(s)$.
 
 \begin{notation}
 Let $\mathbf{T_{small}(s)}:=\{u \in V_{small} : u$ is not covered by
@@ -974,7 +944,7 @@
 Being aware of Claim~\ref{claim:claimCoverFromLeft}, the
 task is not to maintain the candidate set, but to generate the
 candidate nodes in $G_{large}$ for a given node $u\in V_{small}$.  In
-case of an expanding problem type and $M$ mapping, if a node $v\in
+case of any of the three problem type and a mapping $M$, if a node $v\in
 V_{large}$ is a potential pair of $u\in V_{small}$, then $\forall
 u'\in V_{small} : (u,u')\in
 E_{small}\ and\ u'\ is\ covered\ by\ M\ \Rightarrow (v,Pair(M,u'))\in