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 \journal{Discrete Applied Mathematics}

 \begin{document}

 \begin{frontmatter}

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 \title{VF2++ --- An Improved Subgraph Isomorphism Algorithm}

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 \author[egres,elte]{Alp{\'a}r J{\"u}ttner}

 \ead{alpar@cs.elte.hu}

 \author[elte]{P{\'e}ter Madarasi}

 \ead{madarasip@caesar.elte.hu}

 \address[egres]{MTA-ELTE Egerv{\'a}ry Research Group, Budapest, Hungary.}

 \address[elte]{Department of Operations Research, E{\"o}tv{\"o}s Lor{\'a}nd University, Budapest, Hungary.}

 \begin{abstract}

   This paper presents a largely improved version of the VF2 algorithm

   for the \emph{Subgraph Isomorphism Problem}. The improvements are

   twofold. Firstly, it is based on a new approach for determining the

   matching order of the nodes, and secondly, more efficient -

   nevertheless easier to compute - cutting rules significantly

   reducing the search space are applied.

   In addition to the usual \emph{Subgraph Isomorphism Problem}, the paper also

   presents specialized algorithms for the \emph{Induced Subgraph

     Isomorphism} and for the \emph{Graph Isomorphism Problems}.

   Finally, an extensive experimental evaluation is provided using a

   wide range of inputs, including both real-life biological and

   chemical datasets and standard randomly generated graph series. The

   results show major and consistent running time improvements over the

   other known methods.

   The C++ implementations of the algorithms are available open-source as 

   part of the LEMON graph and network optimization library.

 \end{abstract}

 \begin{keyword}

   Computational Biology, Subgraph Isomorphism Problem

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 \end{keyword}

 \end{frontmatter}

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 %% main text

 \section{Introduction}

 \label{sec:intro}

 In the last decades, combinatorial structures, and especially graphs

 have been considered with ever increasing interest, and applied to the

 solution of several new and revised questions.  The expressiveness,

 the simplicity and the studiedness of graphs make them practical for

 modelling and appear constantly in several seemingly independent

 fields, such as bioinformatics and chemistry.

 Complex biological systems arise from the interaction and cooperation

 of plenty of molecular components. Getting acquainted with such

 systems at the molecular level is of primary importance, since

 protein-protein interaction, DNA-protein interaction, metabolic

 interaction, transcription factor binding, neuronal networks, and

 hormone signalling networks can be understood this way.

 Many chemical and biological structures can easily be modelled

 as graphs, for instance, a molecular structure can be

 considered as a graph, whose nodes correspond to atoms and whose

 edges to chemical bonds. The similarity and dissimilarity of

 objects corresponding to nodes are incorporated to the model

 by \emph{node labels}. Understanding such networks basically

 requires finding specific subgraphs, thus it calls for efficient

 graph matching algorithms.

 Other real-world fields related to some

 variants of graph matching include pattern recognition

 and machine vision~\cite{HorstBunkeApplications}, symbol recognition~\cite{CordellaVentoSymbolRecognition}, and face identification~\cite{JianzhuangYongFaceIdentification}.  \\

 Subgraph and induced subgraph matching problems are known to be

 NP-Complete~\cite{SubgraphNPC}, while the graph isomorphism problem is

 one of the few problems in NP neither known to be in P nor

 NP-Complete. Although polynomial-time isomorphism algorithms are known

 for various graph classes, like trees and planar

 graphs~\cite{PlanarGraphIso}, bounded valence

 graphs~\cite{BondedDegGraphIso}, interval graphs~\cite{IntervalGraphIso}

 or permutation graphs~\cite{PermGraphIso}. Furthermore, an FPT algorithm has also been presented for the coloured hypergraph isomorphism problem in~\cite{ColoredHiperGraphIso}.

 In the following, some algorithms based on other approaches are

 summarized, which do not need any restrictions on the graphs. Even though,

 an overall polynomial behaviour is not expectable from such an

 alternative, it may often have good practical performance, in fact,

 it might be the best choice even on a graph class for which polynomial

 algorithm is known.

 The first practically usable approach was due to

 \emph{Ullmann}~\cite{Ullmann}, which is a commonly used algorithm based on depth-first

 search with a complex heuristic for reducing the

 number of visited states. A major problem is its $\Theta(n^3)$ space

 complexity, which makes it impractical in the case of big sparse

 graphs.

 In a recent paper, Ullmann~\cite{UllmannBit} presents an

 improved version of this algorithm based on a bit-vector solution for

 the binary Constraint Satisfaction Problem.

 The \emph{Nauty} algorithm~\cite{Nauty} transforms the two graphs to

 a canonical form before starting to check for isomorphism. It has

 been considered as one of the fastest graph isomorphism algorithms,

 although graph categories were shown in which it takes exponentially

 many steps. This algorithm handles only the graph isomorphism problem.

 The \emph{LAD} algorithm~\cite{Lad} uses a depth-first search

 strategy and formulates the matching as a Constraint Satisfaction

 Problem to prune the search tree. The constraints are that the mapping

 has to be injective and edge-preserving, hence it is possible to

 handle new matching types as well.

 The \emph{RI} algorithm~\cite{RI} and its variations are based on a

 state space representation. After reordering the nodes of the graphs,

 it uses some fast executable heuristic checks without using any

 complex pruning rules. It seems to run really efficiently on graphs

 coming from biology, and won the International Contest on Pattern

 Search in Biological Databases~\cite{Content}.

 Currently, the most commonly used algorithm is the

 \emph{VF2}~\cite{VF2}, the improved version of \emph{VF}~\cite{VF}, which was

 designed for solving pattern matching and computer vision problems,

 and has been one of the best overall algorithms for more than a

 decade. Although, it is not as fast as some of the new specialized algorithms, it is still widely used due to its simplicity and space efficiency. VF2 uses

 a state space representation and checks some conditions in each state

 to prune the search tree.

 Meanwhile, another variant called \emph{VF2~Plus}~\cite{VF2Plus} has

 been published. It is considered to be as efficient as the RI

 algorithm and has a strictly better behaviour on large graphs.  The

 main idea of VF2~Plus is to precompute a heuristic node order of the graph to be embedded, in which VF2 works more efficiently.

 This paper introduces \emph{VF2++}, a new further improved algorithm

 for the graph and (induced) subgraph isomorphism problems, which uses

 efficient cutting rules and determines a node order in which VF2 runs

 significantly faster on practical inputs.

 The rest of the paper is structured as

 follows. Section~\ref{sec:ProbStat} defines the exact problems to be

 solved, Section~\ref{sec:VF2Alg} provides a description of VF2. Based

 on that, Section~\ref{sec:VF2ppAlg} introduces VF2++. Some technical

 details necessary for an efficient implementation are discussed in

 Section~\ref{sec:VF2ppImpl}.  Finally, Section~\ref{sec:ExpRes}

 provides a detailed experimental evaluation of VF2++ and its comparison

 to the state-of-the-art algorithm.

 It must also be mentioned that the C++ implementations of the

 algorithms have been made available for evaluation and use under an

 open-source license as a part of LEMON~\cite{LEMON} graph

 library.

 \section{Problem Statement}\label{sec:ProbStat}

 This section provides a formal description of the problems to be

 solved.

 \subsection{Definitions}

 Throughout the paper $G_{1}=(V_{1}, E_{1})$ and

 $G_{2}=(V_{2}, E_{2})$ denote two undirected graphs.

 \begin{definition}

 $\mathcal{L}: (V_{1}\cup V_{2}) \longrightarrow K$ is a \textbf{node

     label function}, where K is an arbitrary set. The elements in K

   are the \textbf{node labels}. Two nodes, u and v are said to be

   \textbf{equivalent} if $\mathcal{L}(u)=\mathcal{L}(v)$.

 \end{definition}

 For the sake of simplicity, in this paper the graph, subgraph and induced subgraph isomorphisms are defined in a more general way.

 \begin{definition}\label{sec:ismorphic}

 $G_{1}$ and $G_{2}$ are \textbf{isomorphic} (by the node label $\mathcal{L}$) if $\exists \mathfrak{m}:

   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\\

 $\forall u,v\in{V_{1}} : (u,v)\in{E_{1}} \Leftrightarrow (\mathfrak{m}(u),\mathfrak{m}(v))\in{E_{2}}$

 \end{center}

 \end{definition}

 \begin{definition}

 $G_{1}$ is a \textbf{subgraph} of $G_{2}$ (by the node label $\mathcal{L}$) if $\exists \mathfrak{m}:

   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\\

 $\forall u,v \in{V_{1}} : (u,v)\in{E_{1}} \Rightarrow (\mathfrak{m}(u),\mathfrak{m}(v))\in E_{2}$

 \end{center}

 \end{definition}

 \begin{definition} 

 $G_{1}$ is an \textbf{induced subgraph} of $G_{2}$ (by the node label $\mathcal{L}$) if $\exists

   \mathfrak{m}: 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

 $\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}

 \subsection{Common problems}\label{sec:CommProb}

 The focus of this paper is on the following problems appearing in many applications.

 The \textbf{subgraph isomorphism problem} is the following: is

 $G_{1}$ isomorphic to any subgraph of $G_{2}$ by a given node

 label?

 The \textbf{induced subgraph isomorphism problem} asks the same about the

 existence of an induced subgraph.

 The \textbf{graph isomorphism problem} can be defined as induced

 subgraph matching problem where the sizes of the two graphs are equal.

 In addition, one may want to find a \textbf{single} embedding or \textbf{enumerate} all of them.

 \section{The VF2 Algorithm}\label{sec:VF2Alg}

 This algorithm is the basis of both the VF2++ and the VF2~Plus.  VF2

 is able to handle all the variations mentioned in Section~\ref{sec:CommProb}.  Although it can also handle directed graphs,

 for the sake of simplicity, only the undirected case will be

 discussed.

 \subsection{Common notations}

 \indent Assume $G_{1}$ is searched in $G_{2}$.  The following

 definitions and notations will be used throughout this paper.

 \begin{definition}

 An injection $\mathfrak{m} : D \longrightarrow V_2$ is called (partial) \textbf{mapping}, where $D\subseteq V_1$.

 \end{definition}

 \begin{notation}

 $\mathfrak{D}(f)$ and $\mathfrak{R}(f)$ denote the domain and the range of a function $f$, respectively.

 \end{notation}

 \begin{definition}

 Mapping $\mathfrak{m}$ \textbf{covers} a node $u\in V_1\cup V_2$ if $u\in \mathfrak{D}(\mathfrak{m})\cup \mathfrak{R}(\mathfrak{m})$.

 \end{definition}

 \begin{definition}

 A mapping $\mathfrak{m}$ is $\mathbf{whole\ mapping}$ if $\mathfrak{m}$ covers all the

 nodes of $V_{1}$, i.e. $\mathfrak{D}(\mathfrak{m})=V_1$.

 \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 $\forall w\in \mathfrak{D}(\mathfrak{m}) : f(w)=\mathfrak{m}(w)$ 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}

 Throughout the paper, $\mathbf{PT}$ denotes a generic problem type

 which can be substituted by any of the $\mathbf{SUB}$, $\mathbf{IND}$

 and $\mathbf{ISO}$ problems, which stand for the the problems mentioned in Section~\ref{sec:CommProb} respectively.

 \end{notation}

 \begin{definition}

 Let $\mathfrak{m}$ be a mapping. The \textbf{consistency function for } $\mathbf{PT}$ is a logical function, for which $\mathbf{Cons_{PT}}(\mathfrak{m})$ is true if and only if $\mathfrak{m}$ satisfies the requirements of $\mathbf{PT}$ considering the subgraphs of $G_{1}$ and $G_{2}$ induced by $\mathfrak{D}(\mathfrak{m})$ and $\mathfrak{R}(\mathfrak{m})$, respectively.

 %$\mathbf{Cons_{PT}}(\mathfrak{m})$ is true if there exists a whole mapping $w$ satisfying the requirements of $PT$, for which $\mathfrak{m}$ is exactly $w$ restricted to $\mathfrak{D}(\mathfrak{m})$.

 \end{definition}

 \begin{definition} 

 Let $\mathfrak{m}$ be a mapping. A logical function $\mathbf{Cut_{PT}}$ is a

 \textbf{cutting function for } $\mathbf{PT}$ if the following

 holds. $\mathbf{Cut_{PT}(\mathfrak{m})}$ is false if there exists a sequence of extend operations, which results in a whole mapping satisfying the requirements of $PT$.

 \end{definition}

 \begin{definition}

 A mapping $\mathfrak{m}$ is said to be \textbf{consistent mapping by} $\mathbf{PT}$ if

   $Cons_{PT}(\mathfrak{m})$ is true.

 \end{definition}

 $Cons_{PT}$ and $Cut_{PT}$ will often be used in the following form.

 \begin{notation}

 Let $\mathbf{Cons_{PT}(p, \mathfrak{m})}:=Cons_{PT}(extend(\mathfrak{m},p))$, and

 $\mathbf{Cut_{PT}(p, \mathfrak{m})}:=Cut_{PT}(extend(\mathfrak{m},p))$, where

 $p\in{V_{1}\backslash\mathfrak{D}(\mathfrak{m}) \!\times\!V_{2}\backslash\mathfrak{R}(\mathfrak{m})}$.

 \end{notation}

 $Cons_{PT}$ will be used to check the consistency of the already

 covered nodes, while $Cut_{PT}$ is for looking ahead to recognize if

 no whole consistent mapping can contain the current mapping.

 \subsection{Overview of the algorithm}

 VF2 begins with an empty mapping and gradually extends it with respect to the consistency and cutting functions until a whole mapping is reached.

 Algorithm~\ref{alg:VF2Pseu} is a high-level description of

 the VF2 algorithm. Each state of the matching process can

 be associated with a mapping $\mathfrak{m}$. The initial state

 is associated with a mapping $\mathfrak{m}$, for which

 $\mathfrak{D}(\mathfrak{m})=\emptyset$, i.e. it starts with an empty mapping.

 \begin{algorithm}

 \algtext*{EndIf}%ne nyomtasson end if-et

 \algtext*{EndFor}%ne

 \algtext*{EndProcedure}%ne nyomtasson ..

 \caption{\hspace{0.5cm}$A\ high\ level\ description\ of\ VF2$}\label{alg:VF2Pseu}

 \begin{algorithmic}[1]

 \Procedure{VF2}{Mapping $\mathfrak{m}$, ProblemType $PT$}

   \If{$\mathfrak{m}$ covers

     $V_{1}$} \State Output($\mathfrak{m}$)

   \Else

   \State Compute the set $P_\mathfrak{m}$ of the candidate pairs for extending $\mathfrak{m}$ \ForAll{$p\in{P_\mathfrak{m}}$} \If{Cons$_{PT}$($p,\mathfrak{m}$) $\wedge$

     $\neg$Cut$_{PT}$($p,\mathfrak{m}$)}

     \State \textbf{call}

   VF2($extend(\mathfrak{m},p)$, $PT$) \EndIf \EndFor \EndIf \EndProcedure

 \end{algorithmic}

 \end{algorithm}

 For the current mapping $\mathfrak{m}$, the algorithm computes $P_\mathfrak{m}$, the set of

 candidate node pairs for extending the current mapping $\mathfrak{m}$.

 For each pair $p$ in $P_\mathfrak{m}$, $Cons_{PT}(p,\mathfrak{m})$ and

 $Cut_{PT}(p,\mathfrak{m})$ are evaluated. If the former is true and

 the latter is false, the whole process is recursively applied to

 $extend(\mathfrak{m},p)$. Otherwise, $extend(\mathfrak{m},p)$ is not consistent by $PT$, or it

 can be proved that $\mathfrak{m}$ can not be extended to a whole mapping.

 In order to make sure of the correctness, see Claim~\ref{claim:consMapps}.

 \begin{claim}\label{claim:consMapps}

 Through consistent mappings, only consistent whole mappings can be

 reached, and all the consistent whole mappings are reachable through

 consistent mappings.

 \end{claim}

 Note that a mapping may be reached in exponentially many different ways, since the

 order of extensions does not influence the nascent mapping.

 However, one may make the following observations.

 %\begin{claim}

 %\label{claim:claimTotOrd}

 %Let $\prec$ be an arbitrary total ordering relation on $V_{1}$.  If

 %the algorithm ignores each $p=(u,v) \in P_\mathfrak{m}$, for which

 %\begin{center}

 %$\exists (\tilde{u},\tilde{v})\in P_\mathfrak{m}: \tilde{u} \prec u$,

 %\end{center}

 %then no mapping can be reached more than once, and each whole mapping %remains reachable.

 %\end{claim}

 \begin{definition}

 A total order $(u_{\sigma(1)},u_{\sigma(2)},..,u_{\sigma(|V_{1}|)})$ of

 $V_{1}$ is \textbf{matching order} if VF2 can cover $u_{\sigma(d)}$ on the $d$-th level for all $d\in\{1,..,|V_{1}|\}$.

 \end{definition}

 \begin{claim}

 \label{claim:claimTotOrd}

 If VF2 is prescribed to cover the nodes of $G_1$ according to a matching order, then no mapping can be reached more than once and each whole mapping remains reachable.

 \end{claim}

 Note that the cornerstone of the improvements to VF2 is to choose a proper

 matching order.

 \subsection{The candidate set}

 \label{candidateComputingVF2}

 Let $P_\mathfrak{m}$ be the set of the candidate pairs for inclusion in $\mathfrak{m}$.

 \begin{notation}

 Let $\mathbf{T_{1}(\mathfrak{m})}:=\{u \in V_{1}\backslash\mathfrak{D}(\mathfrak{m}) : \exists \tilde{u}\in{\mathfrak{D}(\mathfrak{m}): (u,\tilde{u})\in E_{1}}\}$, and

  $\mathbf{T_{2}(\mathfrak{m})} := \{v \in V_{2}\backslash\mathfrak{R}(\mathfrak{m}) : \exists\tilde{v}\in{\mathfrak{R}(\mathfrak{m}):(v,\tilde{v})\in E_{2}}\}$.

 \end{notation}

 The set $P_\mathfrak{m}$ contains the pairs of uncovered neighbours of covered

 nodes, and if there is not such a node pair, all the pairs containing

 two uncovered nodes are added. Formally, let

 \[

  P_\mathfrak{m}\!=\!

   \begin{cases} 

    T_{1}(\mathfrak{m})\times T_{2}(\mathfrak{m})&\hspace{-0.15cm}\text{if }

    T_{1}(\mathfrak{m})\!\neq\!\emptyset\ \text{and }T_{2}(\mathfrak{m})\!\neq

    \emptyset,\\ (V_{1}\!\setminus\!\mathfrak{D}(\mathfrak{m}))\!\times\!(V_{2}\!\setminus\!\mathfrak{R}(\mathfrak{m}))

    &\hspace{-0.15cm}\text{otherwise}.

   \end{cases}

 \]

 \subsection{Consistency}

 Let $p=(u,v)\in V_{1}\times V_{2}$, and suppose $\mathfrak{m}$ is a consistent mapping by

 $PT$. $Cons_{PT}(p,\mathfrak{m})$ checks whether

 adding pair $p$ into $\mathfrak{m}$ leads to a consistent mapping by $PT$.

 For example, the consistency function of the induced subgraph isomorphism problem is the following.

 \begin{notation}

 Let $\mathbf{\Gamma_{1} (u)}:=\{\tilde{u}\in V_{1} :

 (u,\tilde{u})\in E_{1}\}$, and $\mathbf{\Gamma_{2}

   (v)}:=\{\tilde{v}\in V_{2} : (v,\tilde{v})\in E_{2}\}$, where $u\in V_{1}$ and $v\in V_{2}$. That is, $\mathbf{\Gamma_{i} (w)}$ denotes the set of neighbours of node $w$ in $G_i$ $(i=1,2)$.

 \end{notation}

 \begin{claim}

 $extend(\mathfrak{m},(u,v))$ is a consistent mapping by $IND$ if and only if $\mathfrak{m}$ is consistent and $(\forall \tilde{u}\in \mathfrak{D}(\mathfrak{m}): (u,\tilde{u})\in E_{1}

 \Leftrightarrow (v,\mathfrak{m}(\tilde{u}))\in E_{2})$. 

 \end{claim}

 The following formulation gives an efficient way of calculating $Cons_{IND}$.

 \begin{claim}

 $Cons_{IND}((u,v),\mathfrak{m}):=Cons_{IND}(\mathfrak{m})\wedge\mathcal{L}(u)\!\!=\!\!\mathcal{L}(v)\wedge(\forall \tilde{v}\in \Gamma_{2}(v)\cap\mathfrak{R}(\mathfrak{m}):(u,\mathfrak{m}^{-1}(\tilde{v}))\in E_{1})\wedge

   (\forall \tilde{u}\in \Gamma_{1}(u)

   \cap \mathfrak{D}(\mathfrak{m}):(v,\mathfrak{m}(\tilde{u}))\in E_{2})$ is the consistency function for $IND$.

 \end{claim}

 \subsection{Cutting rules}

 $Cut_{PT}(p,\mathfrak{m})$ is defined by a collection of efficiently

 verifiable conditions. The requirement is that $Cut_{PT}(p,\mathfrak{m})$ can

 be true only if it is impossible to extend $extend(\mathfrak{m},p)$ to a

 whole mapping.

 As an example, a cutting function of induced subgraph isomorphism problem is presented.

 \begin{notation}

 Let $\mathbf{\tilde{T}_{1}}(\mathfrak{m}):=(V_{1}\backslash

 \mathfrak{D}(\mathfrak{m}))\backslash T_{1}(\mathfrak{m})$, and

 \\ $\mathbf{\tilde{T}_{2}}(\mathfrak{m}):=(V_{2}\backslash

 \mathfrak{R}(\mathfrak{m}))\backslash T_{2}(\mathfrak{m})$.

 \end{notation}

 \begin{claim}

 $Cut_{IND}((u,v),\mathfrak{m}):= |\Gamma_{2} (v)\ \cap\ T_{2}(\mathfrak{m})| <

   |\Gamma_{1} (u)\ \cap\ T_{1}(\mathfrak{m})| \vee |\Gamma_{2}(v)\cap

   \tilde{T}_{2}(\mathfrak{m})| < |\Gamma_{1}(u)\cap

   \tilde{T}_{1}(\mathfrak{m})|$ is a cutting function for $IND$.

 \end{claim}

 \section{The VF2++ Algorithm}\label{sec:VF2ppAlg}

 Although any matching order makes the search space of VF2 a

 tree, its choice turns out to dramatically influence the number of

 visited states. The goal is to determine an efficient one as quickly

 as possible.

 The main reason for the superiority of VF2++ over VF2 is twofold. Firstly,

 taking into account the structure and the node labelling of the graph,

 VF2++ determines a matching order in which most of the unfruitful

 branches of the search space can be pruned immediately. Secondly,

 introducing more efficient --- nevertheless still easier to compute

 --- cutting rules reduces the chance of going astray even further.

 In addition to the usual subgraph isomorphism problem, specialized versions

 for induced subgraph and graph isomorphism problems have been

 designed.

 Note that a weaker version of the cutting rules of VF2++ and an efficient

 candidate set calculation method were described in~\cite{VF2Plus}.

 It should be noted that all the methods described in this section are

 extendable to handle directed graphs and edge labels as well.

 The basic ideas and the detailed description of VF2++ are provided in

 the following.\newline

 The goal is to find a matching order in which the algorithm is able to

 recognize inconsistency or prune the infeasible branches on the

 highest levels and goes deep only if it is needed.

 \begin{notation}

 Let $\mathbf{Conn_{H}(u)}:=|\Gamma_{1}(u)\cap H|$, that is the

 number of neighbours of u which are in H, where $u\in V_{1} $ and

 $H\subseteq V_{1}$.

 \end{notation}

 The principal question is the following. Suppose a mapping $\mathfrak{m}$ is

 given. For which node of $T_{1}(\mathfrak{m})$ is the hardest to find a

 consistent pair in $G_{2}$? The more covered neighbours a node in

 $T_{1}(\mathfrak{m})$ has --- i.e. the largest $Conn_{\mathfrak{D}(\mathfrak{m})}$ it has

 ---, the more rarely satisfiable consistency constraints for its pair

 are given.

 Most of the graphs of biological and chemical structures are sparse, thus several nodes in

 $T_{1}(\mathfrak{m})$ may have the same $Conn_{\mathfrak{D}(\mathfrak{m})}$, which makes

 reasonable to define a secondary and a tertiary order between them.

 The observation above proves itself to be as determining, that the

 secondary ordering prefers nodes with the most uncovered neighbours

 among which have the same $Conn_{\mathfrak{D}(\mathfrak{m})}$ to increase

 $Conn_{\mathfrak{D}(\mathfrak{m})}$ of uncovered nodes as much, as possible.  The tertiary ordering prefers nodes having the rarest uncovered labels in $G_2$.

 Note that the secondary ordering is the same as ordering by degrees,

 which is a static data in front of the above used.

 These rules can easily result in a matching order which contains the

 nodes of a long path successively, whose nodes may have low $Conn$ and

 is easily matchable into $G_{2}$. To try to avoid that, a Breadth-first-search order is used, and on each of its levels, the ordering procedure described above is applied.

 \newline

 In the following, some examples on which the VF2 may be slow are

 described, although they are easily solvable by using a proper

 matching order.

 \begin{example}

 Suppose $G_{1}$ can be mapped into $G_{2}$ in many ways

 without node labels. Let $u\in V_{1}$ and $v\in V_{2}$.

 \newline

 $\mathcal{L}(u):=black$

 \newline

 $\mathcal{L}(v):=black$

 \newline

 $\mathcal{L}(\tilde{u}):=red \ \forall \tilde{u}\in V_{1}\backslash

 \{u\}$

 \newline

 $\mathcal{L}(\tilde{v}):=red \ \forall \tilde{v}\in V_{2}\backslash

 \{v\}$

 Now, any mapping by $\mathcal{L}$ must contain $(u,v)$, since

 $u$ is black and no node in $V_{2}$ has a black label except

 $v$. If unfortunately $u$ were the last node which will get covered,

 VF2 would check only in the last steps, whether $u$ can be matched to

 $v$.

 However, had $u$ been the first matched node, u would have been

 matched immediately to v, so all the mappings would have been

 precluded in which node labels can not correspond.

 \end{example}

 \begin{example}

 Suppose there is no node label given, and $G_{1}$ is a small graph that can not be mapped into $G_{2}$ and $u\in V_{1}$.

 \newline

 Let $G'_{1}:=(V_{1}\cup

 \{u'_{1},u'_{2},..,u'_{k}\},E_{1}\cup

 \{(u,u'_{1}),(u'_{1},u'_{2}),..,(u'_{k-1},u'_{k})\})$, that is,

 $G'_{1}$ is $G_{1}\cup \{ a\ k$ long path, which is disjoint

 from $G_{1}$ and one of its starting points is connected to $u\in

 V_{1}\}$.

 If unfortunately the nodes of the path were the first $k$ nodes in the

 matching order, the algorithm would iterate through all the possible k

 long paths in $G_{2}$, and it would recognize that no path can be

 extended to $G'_{1}$.

 \newline

 However, had it started by the matching of $G_{1}$, it would not

 have matched any nodes of the path.

 \end{example}

 These examples may look artificial, but the same problems also appear

 in real-world instances, even though in a less obvious way.

 %\subsection{Preparations}

 %\begin{claim}

 %\label{claim:claimCoverFromLeft}

 %The total ordering relation uniquely determines a node order, in which

 %the nodes of $V_{1}$ will be covered by VF2. From the point of

 %view of the matching procedure, this means, that always the same node

 %of $G_{1}$ will be covered on the $d$-th level.

 %\end{claim}

 %\begin{definition}

 %An order $(u_{\sigma(1)},u_{\sigma(2)},..,u_{\sigma(|V_{1}|)})$ of

 %$V_{1}$ is \textbf{matching order} if there exists $\prec$ total

 %ordering relation, s.t. the VF2 with $\prec$ on the d-th level finds

 %pair for $u_{\sigma(d)}$ for all $d\in\{1,..,|V_{1}|\}$.

 %\end{definition}

 %\begin{claim}\label{claim:MOclaim}

 %A total ordering is matching order iff the nodes of every component

 %form an interval in the node sequence, and every node connects to a

 %previous node in its component except the first node of each component.

 %\end{claim}

 %In summary, a total ordering always uniquely determines a matching

 %order, and every matching order can be determined by a total ordering,

 %however, more than one different total orderings may determine the

 %same matching order.

 \subsection{Matching order}

 \begin{notation}

 Let \textbf{F$_\mathcal{M}$(l)}$:=|\{v\in V_{2} :

 l=\mathcal{L}(v)\}|-|\{u\in \mathcal{M} : l=\mathcal{L}(u)\}|$,

 where $l$ is a label and $\mathcal{M}\subseteq V_{1}$.

 \end{notation}

 \begin{definition}Let $\mathbf{arg\ max}_{f}(S) :=\{u\in S : f(u)=max_{v\in S}\{f(v)\}\}$ and $\mathbf{arg\ min}_{f}(S) := arg\ max_{(-f)}(S)$, where $S$ is a finite set and $f:S\longrightarrow \mathbb{R}$.

 \end{definition}

 \begin{notation}

 Let $deg(v)$ denote the degree of node $v$.

 \end{notation}

 \begin{algorithm}[H]

 \algtext*{EndIf}

 \algtext*{EndProcedure}

 \algtext*{EndWhile}

 \algtext*{EndFor}

 \caption{\hspace{0.5cm}$The\ method\ of\ VF2++\ for\ determining\ the\ node\ order$}\label{alg:VF2PPPseu}

 \begin{algorithmic}[1]

 \Procedure{VF2++order}{} \State $\mathcal{M}$ := $\emptyset$

 \Comment{matching order} \While{$V_{1}\backslash \mathcal{M}

   \neq\emptyset$} \State $r\in$ arg max$_{deg}$ (arg

 min$_{F_\mathcal{M}\circ \mathcal{L}}(V_{1}\backslash

 \mathcal{M})$)\label{alg:findMin} \State Compute $T$, a BFS tree with

 root node $r$.  \For{$d=0,1,...,depth(T)$} \State $V_d$:=nodes of the

 $d$-th level \State Process $V_d$ \Comment{See Algorithm~\ref{alg:VF2PPProcess1}} \EndFor

 \EndWhile \EndProcedure

 \end{algorithmic}

 \end{algorithm}

 \begin{algorithm}

 \algtext*{EndIf}

 \algtext*{EndProcedure}%ne nyomtasson ..

 \algtext*{EndWhile}

 \caption{\hspace{.5cm}$The\ method\ for\ processing\ a\ level\ of\ the\ BFS\ tree$}\label{alg:VF2PPProcess1}

 \begin{algorithmic}[1]

 \Procedure{VF2++ProcessLevel}{$V_{d}$} \While{$V_d\neq\emptyset$}

 \State $m\in$ arg min$_{F_\mathcal{M}\circ\ \mathcal{L}}($ arg max$_{deg}($arg

 max$_{Conn_{\mathcal{M}}}(V_{d})))$ \State $V_d:=V_d\backslash m$

 \State Append node $m$ to the end of $\mathcal{M}$ \State Refresh

 $F_\mathcal{M}$ \EndWhile \EndProcedure

 \end{algorithmic}

 \end{algorithm}

 Algorithm~\ref{alg:VF2PPPseu} is a high-level description of the

 matching order procedure of VF2++. It computes a BFS tree for each

 component in ascending order of their rarest node labels and largest $deg$,

 whose root vertex is the minimal node of its component. Algorithm~\ref{alg:VF2PPProcess1} is a method to process a level of the BFS tree, which appends the nodes of the current level in descending

 lexicographic order by $(Conn_{\mathcal{M}},deg,-F_\mathcal{M})$ separately

 to $\mathcal{M}$, and refreshes $F_\mathcal{M}$ immediately.

 \begin{claim}

 Algorithm~\ref{alg:VF2PPPseu} provides a matching order.

 \end{claim}

 \subsection{Cutting rules}

 \label{VF2PPCuttingRules}

 This section presents the cutting rules of VF2++, which are improved by using extra information coming from the node labels.

 \begin{notation}

 Let $\mathbf{\Gamma_{1}^{l}(u)}:=\{\tilde{u} : \mathcal{L}(\tilde{u})=l

 \wedge \tilde{u}\in \Gamma_{1} (u)\}$ and

 $\mathbf{\Gamma_{2}^{l}(v)}:=\{\tilde{v} : \mathcal{L}(\tilde{v})=l \wedge

 \tilde{v}\in \Gamma_{2} (v)\}$, where $u\in V_{1}$, $v\in

 V_{2}$ and $l$ is a label.

 \end{notation}

 \begin{claim}[Cutting function for ISO]

 \[LabCut_{ISO}((u,v),\mathfrak{m}):=\bigvee_{l\ is\ label}|\Gamma_{2}^{l} (v) \cap T_{2}(\mathfrak{m})|\!\neq\!|\Gamma_{1}^{l}(u)\cap T_{1}(\mathfrak{m})|\  \vee\]\[\bigvee_{l\ is\ label} \newline |\Gamma_{2}^{l}(v)\cap \tilde{T}_{2}(\mathfrak{m})| \neq |\Gamma_{1}^{l}(u)\cap \tilde{T}_{1}(\mathfrak{m})|\] is a cutting function for ISO.

 \end{claim}

 \begin{claim}[Cutting function for IND]

 \[LabCut_{IND}((u,v),\mathfrak{m}):=\bigvee_{l\ is\ label}|\Gamma_{2}^{l} (v) \cap T_{2}(\mathfrak{m})|\!<\!|\Gamma_{1}^{l}(u)\cap T_{1}(\mathfrak{m})|\ \vee\]\[\bigvee_{l\ is\ label} \newline |\Gamma_{2}^{l}(v)\cap \tilde{T}_{2}(\mathfrak{m})| < |\Gamma_{1}^{l}(u)\cap \tilde{T}_{1}(\mathfrak{m})|\] is a cutting function for IND.

 \end{claim}

 \begin{claim}[Cutting function for SUB]

 \[LabCut_{SU\!B}((u,v),\mathfrak{m}):=\bigvee_{l\ is\ label}|\Gamma_{2}^{l} (v) \cap T_{2}(\mathfrak{m})|\!<\!|\Gamma_{1}^{l}(u)\cap T_{1}(\mathfrak{m})|\] is a cutting function for SUB.

 \end{claim}

 \section{Implementation details}\label{sec:VF2ppImpl}

 This section provides a detailed summary of an efficient

 implementation of VF2++.

 \begin{notation}

 Let $\Delta_1$ and $\Delta_2$ denote the largest degree in $G_1$ and $G_2$, respectively, and let $\Delta=\max\{\Delta_1,\Delta_2\}$.

 \end{notation}

 \subsection{Storing a mapping}

 After fixing an arbitrary node order ($u_0, u_1, ..,

 u_{|V_{1}|-1}$) of $G_{1}$, an array $M$ can be used to store

 the current mapping in the following way.

 \[

  M[i] =

   \begin{cases} 

    v & if\ (u_i,v)\ is\ in\ the\ mapping\\ INV\!ALI\!D &

    if\ no\ node\ has\ been\ mapped\ to\ u_i,

   \end{cases}

 \]

 where $i\in\{0,1, ..,|V_{1}|-1\}$, $v\in V_{2}$ and $INV\!ALI\!D$

 means "no node".

 \subsection{Avoiding the recurrence}

 The recursion of Algorithm~\ref{alg:VF2Pseu} can be realized

 as a \textit{while loop}, which has a loop counter $depth$ denoting the

 current depth of the recursion. Fixing a matching order, let $M$

 denote the array storing the current mapping. Observe that

 $M$ is $INV\!ALI\!D$ from index $depth$+1 and not $INV\!ALI\!D$ before

 $depth$. $M[depth]$ changes

 while the state is being processed, but the property is held before

 both stepping back to a predecessor state and exploring a successor

 state.

 The necessary part of the candidate set is easily maintainable or

 computable by following

 the steps described in Section~\ref{candidateComputingVF2}. A much faster method

 has been designed for biological and sparse graphs, see the next

 section for details.

 \subsection{Calculating the candidates for a node}

 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 $\mathfrak{m}$, if a node $v\in

 V_{2}$ is a potential pair of $u\in V_{1}$, then $\forall

 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 $v$, i.e. selecting arbitrarily a covered neighbour $u'$ of $u$, all of the admissible candidates for $u$ are among the neighbours of $\mathfrak{m}(u')$.

 Having said that, an algorithm running in $\Theta(\Delta_2)$ time is

 describable if there exists a covered node in the component containing

 $u$, and a linear one otherwise.

 \subsection{Determining the node order}

 This section describes how the node order preprocessing method of

 VF2++ can efficiently be implemented.

 For using lookup tables, the node labels are associated with the

 numbers $\{0,1,..,|K|-1\}$, where $K$ is the set of the labels. It

 enables $F_\mathcal{M}$ to be stored in an array. At first, the node order

 $\mathcal{M}=\emptyset$, so $F_\mathcal{M}[i]$ is the number of nodes

 in $V_{2}$ having label $i$, which is easy to compute in

 $\Theta(|V_{2}|)$ steps.

 Representing $\mathcal{M}\subseteq V_{1}$ as an array of

 size $|V_{1}|$, both the computation of the BFS tree, and processing its levels by Algorithm~\ref{alg:VF2PPProcess1} can be done in-place by swapping nodes.

 \subsection{Cutting rules}

 In Section~\ref{VF2PPCuttingRules}, the cutting rules were

 described using the sets $T_{1}$, $T_{2}$, $\tilde T_{1}$

 and $\tilde T_{2}$, which are dependent on the current mapping. The aim is to check the labelled cutting

 rules of VF2++ in $\Theta(\Delta)$ time.

 Firstly, suppose that these four sets are given in such a way, that

 checking whether a node is in a certain set takes constant time,

 e.g. they are given by their 0-1 characteristic vectors. Let $L$ be an

 initially zero integer lookup table of size $|K|$. After incrementing

 $L[\mathcal{L}(u')]$ for all $u'\in \Gamma_{1}(u) \cap T_{1}(\mathfrak{m})$ and

 decrementing $L[\mathcal{L}(v')]$ for all $v'\in\Gamma_{2} (v) \cap

 T_{2}(\mathfrak{m})$, the first part of the cutting rules is checkable in

 $\Theta(\Delta)$ time by considering the proper signs of $L$. Setting $L$

 to zero takes $\Theta(\Delta)$ time again, which makes it possible to use

 the same table through the whole algorithm. The second part of the

 cutting rules can be verified using the same method with $\tilde

 T_{1}$ and $\tilde T_{2}$ instead of $T_{1}$ and

 $T_{2}$. Thus, the overall time complexity is $\Theta(\Delta)$.

 To maintain the sets $T_{1}$, $T_{2}$, $\tilde T_{1}$

 and $\tilde T_{2}$, two other integer lookup tables storing the number of covered neighbours of the nodes of the two graphs can be used. This representation allows constant-time membership checking, furthermore it is maintainable in $\Theta(\Delta)$ time whenever a node pair is added or subtracted by incrementing

 or decrementing the proper indices. A further improvement is that the

 values of $L[\mathcal{L}(u')]$ in case of checking $u$ are dependent only on

 $u$, i.e. on the current depth of the recursion, so for each $u\in V_{1}$, an array of pairs \textit{(label, number of such labels)} can store $L$. Note that these arrays are at most of size

 $\Delta_1$ if pairs with non-appearing node labels are discarded.

 Using similar techniques, the consistency function can be evaluated in

 $\Theta(\Delta)$ steps, as well.

 \section{Experimental results}\label{sec:ExpRes}

 This section compares the performance of VF2++ and VF2~Plus. According to

 our experience, both algorithms run faster than VF2 with orders of

 magnitude, thus its inclusion was not reasonable.

 The algorithms were implemented in C++ using the open-source

 LEMON graph and network optimization library~\cite{LEMON}. The tests were carried out on a Linux-based system with an Intel i7 X980 3.33 GHz CPU and 6 GB of RAM.

 \subsection{Biological graphs}

 The tests have been executed on a recent biological dataset created

 for the International Contest on Pattern Search in Biological

 Databases~\cite{Content}, which has been constructed of molecule,

 protein and contact map graphs extracted from the Protein Data

 Bank~\cite{ProteinDataBank}.

 The molecule dataset contains small graphs with less than 100 nodes

 and an average degree of less than 3. The protein dataset contains

 graphs having 500-10 000 nodes and an average degree of 4, while the

 contact map dataset contains graphs with 150-800 nodes and an average

 degree of 20.  \\

 In the following, both the induced subgraph and the graph

 isomorphism problems will be examined.

 This dataset provides graph pairs, between which all the induced subgraph isomorphisms have to be found. For runtime results, please see Figure~\ref{fig:bioIND}.

 In an other experiment, the nodes of each graph in the database had been

 shuffled, and an isomorphism between the shuffled and the original

 graph was searched. The running times are shown on Figure~\ref{fig:bioISO}.

 \begin{figure}[H]

 \vspace*{-2cm}

 \hspace*{-1.5cm}

 \begin{subfigure}[b]{0.55\textwidth}

 \begin{figure}[H]

 \begin{tikzpicture}[trim axis left, trim axis right]

 \begin{axis}[title=Molecules IND,xlabel={$|V_2|$},ylabel={time (ms)},legend entries={VF2 Plus,VF2++},grid

 =major,mark size=1.2pt, legend style={at={(0,1)},anchor=north

   west},scaled x ticks = false,x tick label style={/pgf/number

   format/1000 sep = \kern 0.08em},y tick label style={/pgf/number

   format/1000 sep = \kern 0.08em}]

 %\addplot+[only marks] table {proteinsOrig.txt};

 \addplot table {Orig/Molecules.32.txt}; \addplot[mark=triangle*,mark

   size=1.8pt,color=red] table {VF2PPLabel/Molecules.32.txt};

 \end{axis}

 \end{tikzpicture}

 \caption{In the case of molecules, the algorithms have

   similar behaviour, but VF2++ is almost two times faster even on such

   small graphs.} \label{fig:INDMolecule}

 \end{figure}

 \end{subfigure}

 \hspace*{1.5cm}

 \begin{subfigure}[b]{0.55\textwidth}

 \begin{figure}[H]

 \begin{tikzpicture}[trim axis left, trim axis right]

 \begin{axis}[title=Contact maps IND,xlabel={$|V_2|$},ylabel={time (ms)},legend entries={VF2 Plus,VF2++},grid

 =major,mark size=1.2pt, legend style={at={(0,1)},anchor=north

   west},scaled x ticks = false,x tick label style={/pgf/number

   format/1000 sep = \kern 0.08em},y tick label style={/pgf/number

   format/1000 sep = \kern 0.08em}]

 %\addplot+[only marks] table {proteinsOrig.txt};

 \addplot table {Orig/ContactMaps.128.txt};

 \addplot[mark=triangle*,mark size=1.8pt,color=red] table

         {VF2PPLabel/ContactMaps.128.txt};

 \end{axis}

 \end{tikzpicture}

 \caption{On contact maps, VF2++ runs almost in constant time, while VF2

   Plus has a near-linear behaviour.} \label{fig:INDContact}

 \end{figure}

 \end{subfigure}

 \begin{center}

 \vspace*{-0.5cm}

 \begin{subfigure}[b]{0.55\textwidth}

 \begin{figure}[H]

 \begin{tikzpicture}[trim axis left, trim axis right]

   \begin{axis}[title=Proteins IND,xlabel={$|V_2|$},ylabel={time (ms)},legend entries={VF2 Plus,VF2++},grid

   =major,mark size=1.2pt, legend style={at={(0,1)},anchor=north

     west},scaled x ticks = false,x tick label style={/pgf/number

   format/1000 sep = \kern 0.08em},y tick label style={/pgf/number

   format/1000 sep = \kern 0.08em}] %\addplot+[only marks] table

     {proteinsOrig.txt}; \addplot[mark=*,mark size=1.2pt,color=blue]

     table {Orig/Proteins.256.txt}; \addplot[mark=triangle*,mark

       size=1.8pt,color=red] table {VF2PPLabel/Proteins.256.txt};

   \end{axis}

   \end{tikzpicture}

 \caption{Both of the algorithms have linear behaviour on protein

   graphs. VF2++ is more than 10 times faster than VF2

   Plus.} \label{fig:INDProt}

 \end{figure}

 \end{subfigure}

 \end{center}

 \vspace*{-0.5cm}

 \caption{\normalsize{Induced subgraph isomorphism problem on biological graphs}}\label{fig:bioIND}

 \end{figure}

 \begin{figure}[H]

 \vspace*{-2cm}

 \hspace*{-1.5cm}

 \begin{subfigure}[b]{0.55\textwidth}

 \begin{figure}[H]

 \begin{tikzpicture}[trim axis left, trim axis right]

 \begin{axis}[title=Molecules ISO,xlabel={$|V_2|$},ylabel={time (ms)},legend entries={VF2 Plus,VF2++},grid

 =major,mark size=1.2pt, legend style={at={(0,1)},anchor=north

   west},scaled x ticks = false,x tick label style={/pgf/number

   format/1000 sep = \kern 0.08em},y tick label style={/pgf/number

   format/1000 sep = \kern 0.08em}]

 %\addplot+[only marks] table {proteinsOrig.txt};

 \addplot table {Orig/moleculesIso.txt}; \addplot[mark=triangle*,mark

   size=1.8pt,color=red] table {VF2PPLabel/moleculesIso.txt};

 \end{axis}

 \end{tikzpicture}

 \caption{The results are close to each other on contact maps, but VF2++ seems to be slightly faster as the number of nodes increases.

   }\label{fig:ISOMolecule}

 \end{figure}

 \end{subfigure}

 \hspace*{1.5cm}

 \begin{subfigure}[b]{0.55\textwidth}

 \begin{figure}[H]

 \begin{tikzpicture}[trim axis left, trim axis right]

 \begin{axis}[title=Contact maps ISO,xlabel={$|V_2|$},ylabel={time (ms)},legend entries={VF2 Plus,VF2++},grid

 =major,mark size=1.2pt, legend style={at={(0,1)},anchor=north

   west},scaled x ticks = false,x tick label style={/pgf/number

   format/1000 sep = \kern 0.08em},y tick label style={/pgf/number

   format/1000 sep = \kern 0.08em}]

 %\addplot+[only marks] table {proteinsOrig.txt};

 \addplot table {Orig/contactMapsIso.txt}; \addplot[mark=triangle*,mark

   size=1.8pt,color=red] table {VF2PPLabel/contactMapsIso.txt};

 \end{axis}

 \end{tikzpicture}

 \caption{In the case of molecules, there is no significant

   difference, but VF2++ performs consistently better.}\label{fig:ISOContact}

 \end{figure}

 \end{subfigure}

 \begin{center}

 \vspace*{-0.5cm}

 \begin{subfigure}[b]{0.55\textwidth}

 \begin{figure}[H]

 \begin{tikzpicture}[trim axis left, trim axis right]

 \begin{axis}[title=Proteins ISO,xlabel={$|V_2|$},ylabel={time (ms)},legend entries={VF2 Plus,VF2++},grid

 =major,mark size=1.2pt, legend style={at={(0,1)},anchor=north

   west},scaled x ticks = false,x tick label style={/pgf/number

   format/1000 sep = \kern 0.08em},y tick label style={/pgf/number

   format/1000 sep = \kern 0.08em}]

 %\addplot+[only marks] table {proteinsOrig.txt};

 \addplot table {Orig/proteinsIso.txt}; \addplot[mark=triangle*,mark

   size=1.8pt,color=red] table {VF2PPLabel/proteinsIso.txt};

 \end{axis}

 \end{tikzpicture}

 \caption{On protein graphs, VF2~Plus has a super linear time

   complexity, while VF2++ runs in near constant time. The difference

   is about two orders of magnitude on large graphs.}\label{fig:ISOProt}

 \end{figure}

 \end{subfigure}

 \end{center}

 \vspace*{-0.6cm}

 \caption{\normalsize{Graph isomorphism problem on biological graphs}}\label{fig:bioISO}

 \end{figure}

 \subsection{Random graphs}

 This section compares VF2++ with VF2~Plus on random graphs of large

 size. The node labels are uniformly distributed.  Let $\delta$ denote

 the average degree.  For the parameters of problems solved in the

 experiments, please see the top of each chart.

 \subsubsection{Graph isomorphism problem}

 To evaluate the efficiency of the algorithms in the case of graph

 isomorphism problem, random connected graphs of less than 20 000 nodes have been

 considered. Generating a random graph and shuffling its nodes, an

 isomorphism had to be found. Figure~\ref{fig:randISO} shows the runtime results

 on graph sets of various density.

 \begin{figure}

 \vspace*{-1.5cm}

 \hspace*{-1.5cm}

 \begin{subfigure}[b]{0.55\textwidth}

 \begin{center}

 \begin{tikzpicture}

 \begin{axis}[title={Random ISO, $\delta = 5$},width=7.2cm,height=6cm,xlabel={$|V_2|$},ylabel={time (ms)},ylabel near ticks,legend entries={VF2 Plus,VF2++},grid

 =major,mark size=1.2pt, legend style={at={(0,1)},anchor=north

   west},scaled x ticks = false,x tick label style={/pgf/number

   format/1000 sep = \kern 0.08em},y tick label style={/pgf/number

   format/1000 sep = \kern 0.08em}]

 %\addplot+[only marks] table {proteinsOrig.txt};

 \addplot table {randGraph/iso/vf2pIso5_1.txt};

 \addplot[mark=triangle*,mark size=1.8pt,color=red] table

         {randGraph/iso/vf2ppIso5_1.txt};

 \end{axis}

 \end{tikzpicture}

 \end{center}

 \end{subfigure}

 %\hspace{1cm}

 \begin{subfigure}[b]{0.55\textwidth}

 \begin{center}

 \begin{tikzpicture}

 \begin{axis}[title={Random ISO, $\delta = 10$},width=7.2cm,height=6cm,xlabel={$|V_2|$},ylabel={time (ms)},ylabel near ticks,legend entries={VF2 Plus,VF2++},grid

 =major,mark size=1.2pt, legend style={at={(0,1)},anchor=north

   west},scaled x ticks = false,x tick label style={/pgf/number

   format/1000 sep = \kern 0.08em},y tick label style={/pgf/number

   format/1000 sep = \kern 0.08em}]

 %\addplot+[only marks] table {proteinsOrig.txt};

 \addplot table {randGraph/iso/vf2pIso10_1.txt};

 \addplot[mark=triangle*,mark size=1.8pt,color=red] table

         {randGraph/iso/vf2ppIso10_1.txt};

 \end{axis}

 \end{tikzpicture}

 \end{center}

 \end{subfigure}

 %%\hspace{1cm}

 \hspace*{-1.5cm}

 \begin{subfigure}[b]{0.55\textwidth}

 \begin{center}

 \begin{tikzpicture}

 \begin{axis}[title={Random ISO, $\delta = 15$},width=7.2cm,height=6cm,xlabel={$|V_2|$},ylabel={time (ms)},ylabel near ticks,legend entries={VF2 Plus,VF2++},grid

 =major,mark size=1.2pt, legend style={at={(0,1)},anchor=north

   west},scaled x ticks = false,x tick label style={/pgf/number

   format/1000 sep = \kern 0.08em},y tick label style={/pgf/number

   format/1000 sep = \kern 0.08em}]

 %\addplot+[only marks] table {proteinsOrig.txt};

 \addplot table {randGraph/iso/vf2pIso15_1.txt};

 \addplot[mark=triangle*,mark size=1.8pt,color=red] table

         {randGraph/iso/vf2ppIso15_1.txt};

 \end{axis}

 \end{tikzpicture}

 \end{center}

      \end{subfigure}

      \begin{subfigure}[b]{0.55\textwidth}

 \begin{center}

 \begin{tikzpicture}

 \begin{axis}[title={Random ISO, $\delta = 100$},width=7.2cm,height=6cm,xlabel={$|V_2|$},ylabel={time (ms)},ylabel near ticks,legend entries={VF2 Plus,VF2++},grid

 =major,mark size=1.2pt, legend style={at={(0,1)},anchor=north

   west},scaled x ticks = false,x tick label style={/pgf/number

   format/1000 sep = \kern 0.08em},y tick label style={/pgf/number

   format/1000 sep = \kern 0.08em}]

 %\addplot+[only marks] table {proteinsOrig.txt};

 \addplot table {randGraph/iso/vf2pIso100_1.txt};

 \addplot[mark=triangle*,mark size=1.8pt,color=red] table

         {randGraph/iso/vf2ppIso100_1.txt};

 \end{axis}

 \end{tikzpicture}

 \end{center}

 \end{subfigure}

 \vspace*{-0.8cm}

 \caption{Graph isomorphism problem on random graphs

 }\label{fig:randISO}

 \end{figure}

 \subsubsection{Induced subgraph isomorphism problem}

 This section presents a comparison of VF2++ and VF2~Plus in the case

 of induced subgraph isomorphism problem. In addition to the size of graph $G_2$, that of $G_1$ dramatically influences the hardness of

 a given problem too, so the overall picture is provided by examining graphs to be embedded of various size.

 For each chart, a number $0<\rho< 1$ has been fixed, and the following

 has been executed 150 times. Generating a large graph $G_{2}$ of an average degree of $\delta$,

 choose 10 of its induced subgraphs having $\rho|V_{2}|$ nodes,

 and for all the 10 subgraphs find a mapping by using both graph

 matching algorithms.  The $\delta = 5, 10, 35$ and $\rho = 0.05, 0.1,

 0.3, 0.8$ cases have been examined, see

 Figure~\ref{fig:randIND5},~\ref{fig:randIND10}~and~\ref{fig:randIND35}.

 \begin{figure}

 \vspace*{-1.5cm}

 \hspace*{-1.5cm}

 \begin{subfigure}[b]{0.55\textwidth}

 \begin{center}

 \begin{tikzpicture}

 \begin{axis}[title={Random IND, $\delta = 5$, $\rho = 0.05$},width=7.2cm,height=6cm,xlabel={$|V_2|$},ylabel={time (ms)},ylabel near ticks,legend entries={VF2 Plus,VF2++},grid

 =major,mark size=1.2pt, legend style={at={(0,1)},anchor=north

   west},scaled x ticks = false,x tick label style={/pgf/number

   format/1000 sep = \kern 0.08em},y tick label style={/pgf/number

   format/1000 sep = \kern 0.08em}]

 %\addplot+[only marks] table {proteinsOrig.txt};

 \addplot table {randGraph/ind/vf2pInd5_0.05.txt};

 \addplot[mark=triangle*,mark size=1.8pt,color=red] table

         {randGraph/ind/vf2ppInd5_0.05.txt};

 \end{axis}

 \end{tikzpicture}

 \end{center}

      \end{subfigure}

      \begin{subfigure}[b]{0.55\textwidth}

 \begin{center}

 \begin{tikzpicture}

 \begin{axis}[title={Random IND, $\delta = 5$, $\rho = 0.1$},width=7.2cm,height=6cm,xlabel={$|V_2|$},ylabel={time (ms)},ylabel near ticks,legend entries={VF2 Plus,VF2++},grid

 =major,mark size=1.2pt, legend style={at={(0,1)},anchor=north

   west},scaled x ticks = false,x tick label style={/pgf/number

   format/1000 sep = \kern 0.08em},y tick label style={/pgf/number

   format/1000 sep = \kern 0.08em}]

 %\addplot+[only marks] table {proteinsOrig.txt};

 \addplot table {randGraph/ind/vf2pInd5_0.1.txt};

 \addplot[mark=triangle*,mark size=1.8pt,color=red] table

         {randGraph/ind/vf2ppInd5_0.1.txt};

 \end{axis}

 \end{tikzpicture}

 \end{center}

 \end{subfigure}

 \hspace*{-1.5cm}

 \begin{subfigure}[b]{0.55\textwidth}

 \begin{center}

 \begin{tikzpicture}

 \begin{axis}[title={Random IND, $\delta = 5$, $\rho = 0.3$},width=7.2cm,height=6cm,xlabel={$|V_2|$},ylabel={time (ms)},ylabel near ticks,legend entries={VF2 Plus,VF2++},grid

 =major,mark size=1.2pt, legend style={at={(0,1)},anchor=north

   west},scaled x ticks = false,x tick label style={/pgf/number

   format/1000 sep = \kern 0.08em},y tick label style={/pgf/number

   format/1000 sep = \kern 0.08em}]

 %\addplot+[only marks] table {proteinsOrig.txt};

 \addplot table {randGraph/ind/vf2pInd5_0.3.txt};

 \addplot[mark=triangle*,mark size=1.8pt,color=red] table

         {randGraph/ind/vf2ppInd5_0.3.txt};

 \end{axis}

 \end{tikzpicture}

 \end{center}

      \end{subfigure}

      \begin{subfigure}[b]{0.55\textwidth}

 \begin{center}

 \begin{tikzpicture}

 \begin{axis}[title={Random IND, $\delta = 5$, $\rho = 0.8$},width=7.2cm,height=6cm,xlabel={$|V_2|$},ylabel={time (ms)},ylabel near ticks,legend entries={VF2 Plus,VF2++},grid

 =major,mark size=1.2pt, legend style={at={(0,1)},anchor=north

   west},scaled x ticks = false,x tick label style={/pgf/number

   format/1000 sep = \kern 0.08em},y tick label style={/pgf/number

   format/1000 sep = \kern 0.08em}]

 %\addplot+[only marks] table {proteinsOrig.txt};

 \addplot table {randGraph/ind/vf2pInd5_0.8.txt};

 \addplot[mark=triangle*,mark size=1.8pt,color=red] table

         {randGraph/ind/vf2ppInd5_0.8.txt};

 \end{axis}

 \end{tikzpicture}

 \end{center}

 \end{subfigure}

 \vspace*{-0.8cm}

 \caption{Induced subgraph isomorphism problem on random graphs having an average degree of

   5}\label{fig:randIND5}

 \end{figure}

 \begin{figure}

 \vspace*{-1.5cm}

 \hspace*{-1.5cm}

 \begin{subfigure}[b]{0.55\textwidth}

 \begin{center}

 \hspace*{-0.5cm}

 \begin{tikzpicture}

 \begin{axis}[title={Random IND, $\delta = 10$, $\rho = 0.05$},width=7.2cm,height=6cm,xlabel={$|V_2|$},ylabel={time (ms)},ylabel near ticks,legend entries={VF2 Plus,VF2++},grid

 =major,mark size=1.2pt, legend style={at={(0,1)},anchor=north

   west},scaled x ticks = false,x tick label style={/pgf/number

   format/1000 sep = \kern 0.08em},y tick label style={/pgf/number

   format/1000 sep = \kern 0.08em}]

 %\addplot+[only marks] table {proteinsOrig.txt};

 \addplot table {randGraph/ind/vf2pInd10_0.05.txt};

 \addplot[mark=triangle*,mark size=1.8pt,color=red] table

         {randGraph/ind/vf2ppInd10_0.05.txt};

 \end{axis}

 \end{tikzpicture}

 \end{center}

      \end{subfigure}

      \begin{subfigure}[b]{0.55\textwidth}

 \begin{center}

      \hspace*{-0.5cm}

 \begin{tikzpicture}

 \begin{axis}[title={Random IND, $\delta = 10$, $\rho = 0.1$},width=7.2cm,height=6cm,xlabel={$|V_2|$},ylabel={time (ms)},ylabel near ticks,legend entries={VF2 Plus,VF2++},grid

 =major,mark size=1.2pt, legend style={at={(0,1)},anchor=north

   west},scaled x ticks = false,x tick label style={/pgf/number

   format/1000 sep = \kern 0.08em},y tick label style={/pgf/number

   format/1000 sep = \kern 0.08em}]

 %\addplot+[only marks] table {proteinsOrig.txt};

 \addplot table {randGraph/ind/vf2pInd10_0.1.txt};

 \addplot[mark=triangle*,mark size=1.8pt,color=red] table

         {randGraph/ind/vf2ppInd10_0.1.txt};

 \end{axis}

 \end{tikzpicture}

 \end{center}

 \end{subfigure}

 \hspace*{-1.5cm}

 \begin{subfigure}[b]{0.55\textwidth}

 \begin{center}

 \begin{tikzpicture}

 \begin{axis}[title={Random IND, $\delta = 10$, $\rho = 0.3$},width=7.2cm,height=6cm,xlabel={$|V_2|$},ylabel={time (ms)},ylabel near ticks,legend entries={VF2 Plus,VF2++},grid

 =major,mark size=1.2pt, legend style={at={(0,1)},anchor=north

   west},scaled x ticks = false,x tick label style={/pgf/number

   format/1000 sep = \kern 0.08em},y tick label style={/pgf/number

   format/1000 sep = \kern 0.08em}]

 %\addplot+[only marks] table {proteinsOrig.txt};

 \addplot table {randGraph/ind/vf2pInd10_0.3.txt};

 \addplot[mark=triangle*,mark size=1.8pt,color=red] table

         {randGraph/ind/vf2ppInd10_0.3.txt};

 \end{axis}

 \end{tikzpicture}

 \end{center}

      \end{subfigure}

      \begin{subfigure}[b]{0.55\textwidth}

 \begin{center}

 \begin{tikzpicture}

 \begin{axis}[title={Random IND, $\delta = 10$, $\rho = 0.8$},width=7.2cm,height=6cm,xlabel={$|V_2|$},ylabel={time (ms)},ylabel near ticks,legend entries={VF2 Plus,VF2++},grid

 =major,mark size=1.2pt, legend style={at={(0,1)},anchor=north

   west},scaled x ticks = false,x tick label style={/pgf/number

   format/1000 sep = \kern 0.08em},y tick label style={/pgf/number

   format/1000 sep = \kern 0.08em}]

 %\addplot+[only marks] table {proteinsOrig.txt};

 \addplot table {randGraph/ind/vf2pInd10_0.8.txt};

 \addplot[mark=triangle*,mark size=1.8pt,color=red] table

         {randGraph/ind/vf2ppInd10_0.8.txt};

 \end{axis}

 \end{tikzpicture}

 \end{center}

 \end{subfigure}

 \vspace*{-0.8cm}

 \caption{Induced subgraph isomorphism problem on random graphs having an average degree of

   10}\label{fig:randIND10}

 \end{figure}

 \begin{figure}

 \vspace*{-1.5cm}

 \hspace*{-1.5cm}

 \begin{subfigure}[b]{0.55\textwidth}

 \begin{center}

 \begin{tikzpicture}

 \begin{axis}[title={Random IND, $\delta = 35$, $\rho = 0.05$},width=7.2cm,height=6cm,xlabel={$|V_2|$},ylabel={time (ms)},ylabel near ticks,legend entries={VF2 Plus,VF2++},grid

 =major,mark size=1.2pt, legend style={at={(0,1)},anchor=north

   west},scaled x ticks = false,x tick label style={/pgf/number

   format/1000 sep = \kern 0.08em},y tick label style={/pgf/number

   format/1000 sep = \kern 0.08em}]

 %\addplot+[only marks] table {proteinsOrig.txt};

 \addplot table {randGraph/ind/vf2pInd35_0.05.txt};

 \addplot[mark=triangle*,mark size=1.8pt,color=red] table

         {randGraph/ind/vf2ppInd35_0.05.txt};

 \end{axis}

 \end{tikzpicture}

 \end{center}

      \end{subfigure}

      \begin{subfigure}[b]{0.55\textwidth}

 \begin{center}

 \begin{tikzpicture}

 \begin{axis}[title={Random IND, $\delta = 35$, $\rho = 0.1$},width=7.2cm,height=6cm,xlabel={$|V_2|$},ylabel={time (ms)},ylabel near ticks,legend entries={VF2 Plus,VF2++},grid

 =major,mark size=1.2pt, legend style={at={(0,1)},anchor=north

   west},scaled x ticks = false,x tick label style={/pgf/number

   format/1000 sep = \kern 0.08em},y tick label style={/pgf/number

   format/1000 sep = \kern 0.08em}]

 %\addplot+[only marks] table {proteinsOrig.txt};

 \addplot table {randGraph/ind/vf2pInd35_0.1.txt};

 \addplot[mark=triangle*,mark size=1.8pt,color=red] table

         {randGraph/ind/vf2ppInd35_0.1.txt};

 \end{axis}

 \end{tikzpicture}

 \end{center}

 \end{subfigure}

 \hspace*{-1.5cm}

 \begin{subfigure}[b]{0.55\textwidth}

 \begin{center}

 \begin{tikzpicture}

 \begin{axis}[title={Random IND, $\delta = 35$, $\rho = 0.3$},width=7.2cm,height=6cm,xlabel={$|V_2|$},ylabel={time (ms)},ylabel near ticks,legend entries={VF2 Plus,VF2++},grid

 =major,mark size=1.2pt, legend style={at={(0,1)},anchor=north

   west},scaled x ticks = false,x tick label style={/pgf/number

   format/1000 sep = \kern 0.08em},y tick label style={/pgf/number

   format/1000 sep = \kern 0.08em}]

 %\addplot+[only marks] table {proteinsOrig.txt};

 \addplot table {randGraph/ind/vf2pInd35_0.3.txt};

 \addplot[mark=triangle*,mark size=1.8pt,color=red] table

         {randGraph/ind/vf2ppInd35_0.3.txt};

 \end{axis}

 \end{tikzpicture}

 \end{center}

      \end{subfigure}

      \begin{subfigure}[b]{0.55\textwidth}

 \begin{center}

 \begin{tikzpicture}

 \begin{axis}[title={Random IND, $\delta = 35$, $\rho = 0.8$},width=7.2cm,height=6cm,xlabel={$|V_2|$},ylabel={time (ms)},ylabel near ticks,legend entries={VF2 Plus,VF2++},grid

 =major,mark size=1.2pt, legend style={at={(0,1)},anchor=north

   west},scaled x ticks = false,x tick label style={/pgf/number

   format/1000 sep = \kern 0.08em},y tick label style={/pgf/number

   format/1000 sep = \kern 0.08em}]

 %\addplot+[only marks] table {proteinsOrig.txt};

 \addplot table {randGraph/ind/vf2pInd35_0.8.txt};

 \addplot[mark=triangle*,mark size=1.8pt,color=red] table

         {randGraph/ind/vf2ppInd35_0.8.txt};

 \end{axis}

 \end{tikzpicture}

 \end{center}

 \end{subfigure}

 \vspace*{-0.8cm}

 \caption{Induced subgraph isomorphism problem on random graphs having an average degree of

   35}\label{fig:randIND35}

 \end{figure}

 Based on these experiments, VF2++ is faster than VF2~Plus and able to

 handle really large graphs in milliseconds. Note that when $IND$ was

 considered and the graph to be embedded had proportionally few nodes ($\rho =

 0.05$, or $\rho = 0.1$), then VF2~Plus produced some inefficient node

 orders (e.g. see the $\delta=10$ case on

 Figure~\ref{fig:randIND10}). If these instances had been excluded, the

 charts would have looked similarly to the other ones.

 Unsurprisingly, as denser graphs are considered, both VF2++ and VF2

 Plus slow slightly down, but remain practically usable even on graphs

 having 10 000 nodes.

 \section{Conclusion}

 This paper presented VF2++, a new graph matching algorithm based on VF2, and analysed it from a practical viewpoint.

 Recognizing the importance of the node order and determining an

 efficient one, VF2++ is able to match graphs of thousands of nodes in

 near practically linear time including preprocessing. In addition to

 the proper order, VF2++ uses more efficient cutting

 rules which are easy to compute and make the algorithm able to prune

 most of the unfruitful branches without going astray.

 In order to show the efficiency of the new method, it has been

 compared to VF2~Plus~\cite{VF2Plus}, which is the best contemporary algorithm.

 The experiments show that VF2++ consistently outperforms VF2~Plus on

 biological graphs. It seems to be asymptotically faster on protein and

 on contact map graphs in the case of induced subgraph isomorphism problem,

 while in the case of graph isomorphism problem, it has definitely better

 asymptotic behaviour on protein graphs.

 Regarding random sparse graphs, not only has VF2++ proved itself to be

 faster than VF2~Plus, but it also has a practically linear behaviour both

 in the case of induced subgraph and graph isomorphism problems.

 %%%%%%%%%%%%%%%%

 \section*{Acknowledgement} \label{sec:ack}

 %%%%%%%%%%%%%%%%

 This research project was initiated and sponsored by QuantumBio

 Inc.~\cite{QUANTUMBIO}.

 The authors were supported by the Hungarian Scientific Research Fund -

 OTKA, K109240 and by the J\'anos Bolyai Research Fellowship program of

 the Hungarian Academy of Sciences.

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 \bibliographystyle{elsarticle-num} \bibliography{bibliography}

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 %% TeX file.

 %% \begin{thebibliography}{00}

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