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Graph Isomorphism is the prime example of a computational problem with a wide difference between the best known lower and upper bounds on its complexity. There is a significant gap between extant lower and upper bounds for planar graphs as well. We bridge the gap for this natural and important special case by presenting an upper bound that matches the known log-space hardness [JKMT03]. In fact, we show the formally stronger result that planar graph canonization is in log-space. This improves the previously known upper bound of AC1 [MR91]. Our algorithm first constructs the biconnected component tree of a connected planar graph and then refines each biconnected component into a triconnected component tree. The next step is to log-space reduce the biconnected planar graph isomorphism and canonization problems to those for 3-connected planar graphs, which are known to be in log-space by [DLN08]. This is achieved by using the above decomposition, and by making significant modifications to Lindell’s algorithm for tree canonization, along with changes in the space complexity analysis. The reduction from the connected case to the biconnected case requires further new ideas, including a non-trivial case analysis and a group theoretic lemma to bound the number of automorphisms of a colored 3-connected planar graph. This lemma is crucial for the reduction to work in log-space.

Graph isomorphism is a core problem in graph analysis of various application domains. Given two graphs, the graph isomorphism problem is to determine whether there exists an isomorphism between them. As real-world graphs are getting bigger and bigger, applications demand practically fast algorithms that can run on large-scale graphs. However, existing approaches such as graph canonization and subgraph isomorphism show limited performances on large-scale graphs either in time or space. In this paper, we propose a new approach to graph isomorphism, which is the framework of pairwise color refinement and efficient backtracking. The main features of our approach are: (1) pairwise color refinement and binary cell mapping (2) compressed CS (candidate space), and (3) partial failing set, which together lead to a much faster and scalable algorithm for graph isomorphism. Extensive experiments with real-world datasets show that our approach outperforms state-of-the-art algorithms by up to orders of magnitude in terms of running time.

The state-of-the-art tools for practical graph canonization are all based on the individualization-refinement paradigm, and their difference is primarily in the choice of heuristics they include and in the actual tool implementation. It is thus not possible to make a direct comparison of how individual algorithmic ideas affect the performance on different graph classes. We present an algorithmic software framework that facilitates implementation of heuristics as independent extensions to a common core algorithm. It therefore becomes easy to perform a detailed comparison of the performance and behavior of different algorithmic ideas. Implementations are provided of a range of algorithms for tree traversal, target cell selection, and node invariant, including choices from the literature and new variations. The framework readily supports extraction and visualization of detailed data from separate algorithm executions for subsequent analysis and development of new heuristics. Using collections of different graph classes, we investigate the effect of varying the selections of heuristics, often revealing exactly which individual algorithmic choice is responsible for particularly good or bad performance. On several benchmark collections, including a newly proposed class of difficult instances, we additionally find that our implementation performs better than the current state-of-the-art tools.

We consider global dynamics of reaction systems as introduced by Ehrenfeucht and Rozenberg. The dynamics is represented by a directed graph, the so-called transition graph, and two reaction systems are considered equivalent if their corresponding transition graphs are isomorphic. We introduce the notion of a skeleton (a one-out graph) that uniquely defines a directed graph. We provide the necessary and sufficient conditions for two skeletons to define isomorphic graphs. This provides a necessary and sufficient condition for two reactions systems to be equivalent, as well as a characterization of the directed graphs that correspond to the global dynamics of reaction systems., 14 pages, 6 figures

We present and compare various methods to construct efficient QUBO formulations for the Graph Isomorphism Problem—one of a very few problems in NP that is neither known to be solvable in polynomial time nor NP-complete—and two related Subgraph Isomorphism Problems that are NP-hard. Experimental results on two QUBO formulations of the Graph Isomorphism Problem suggest that our direct formulation is more practical than the others with respect to running on the D-Wave architecture.

Graph canonization is the problem of computing a unique representative, a canon, from the isomorphism class of a given graph. This implies that two graphs are isomorphic exactly if their canons are equal. We show that graphs of bounded tree width can be canonized by logarithmic-space (logspace) algorithms. This implies that the isomorphism problem for graphs of bounded tree width can be decided in logspace. In the light of isomorphism for trees being hard for the complexity class logspace, this makes the ubiquitous class of graphs of bounded tree width one of the few classes of graphs for which the complexity of the isomorphism problem has been exactly determined., Comment: 26 pages

Identification of isomorphism among graphs is one of the computationally challenging tasks in computer science that couldn’t be solved in polynomial time. In this paper, we derive a polynomial time algorithm that allows direct comparison between different graph structures to check for graph isomorphism. This paper suggest to represent graphs in a common mathematical space (Symmetric Positive Semi-Definite space), so that two isomorphic graphs always map to the same coordinates in a mathematical space. This kind of mathematical representation is generated based on the neighbourhood influences between nodes of a graph which enhances the graph topological structure at the node level in the form of krylov subspace, in polynomial time. Experiments are conducted using publicly available benchmark graph database. From the simulation, it is observed that the representation recommended in this work acts like a signature for each graph with guaranteed isomorphism. Further the proposed approach tries to identify the molecular structure of any application-specific graphs and categorizes them effectively in a polynomial time inspite of its NP-completeness.

Graph canonization (that solves also the graph isomorphism question) is an old problem that still attracts a lot of attention today, mainly because of the ubiquitous aspect of graph-based structures in computer science applications. This article present the proofs of correctness for the SCOTT algorithm, a graph canonization algorithm designed to provide a Canonical form for general graphs, namely graphs for which vertices and edges are coloured (labelled). These proofs ensure that the three canonical forms provided by SCOTT are valid, namely a canonical adjacency matrix, a canonical rooted tree (or DAG) and a canonical string. In addition, some crude lower and upper complexity bounds are presented and discussed. Finally some empirical evaluation is provided on a difficult synthetic benchmark with some comparison with the state of the art algorithms.

In the field of chemistry, it is interesting to be able to estimate the physicochemical properties of molecules, especially for industrial applications. These are difficult to estimate by physical simulations, as their implementation often present prohibitive time complexity. However, the emergence of data (public or private) opens new perspectives for the treatment of these problems by statistical methods and machine learning. The main difficulty lies in the characterization of molecules: these are more like a network of atoms (in other words a colored graph) than a vector. Unfortunately, statistical modeling methods usually deal with observations encoded as such, hence the need for specific methods able to deal with graphs- encoded observations, called structure-activity relationships. The aim of this thesis is to take advantage of public corpora to learn the best possible representations of these structures, and to transfer this global knowledge to smaller datasets. We adapted methods used in automatic processing of natural languages to achieve this goal. To implement them, more theoretical work was needed, especially on the graph isomorphism problem. The results obtained on classification / regression tasks are at least competitive with the state of the art, and even sometimes better, in particular on restricted data sets, attesting some opportunities for transfer learning in this field.; Dans le domaine de la chimie, il est intéressant de pouvoir estimer des propriétés physico- chimiques de molécules, notamment pour des applications industrielles. Celles-ci sont difficiles à estimer par simulations physique, présentant une complexité temporelle prohibitive. L'émergence des données (publiques ou privées) ouvre toutefois de nouvelles perspectives pour le traitement de ces problèmes par des méthodes statistiques et d'apprentissage automatique. La principale difficulté réside dans la caractérisation des molécules : celles-ci s'apparentent davantage à un réseau d'atomes (autrement dit un graphe coloré) qu'à un vecteur. Or, les méthodes de modélisation statistiques traitent usuellement avec des observations encodées comme telles, d'où la nécessité de méthodes spécifiques, nommées relations structures-activité, traitant des observations encodées sous forme de graphes. Le but de cette thèse est de tirer parti des corpus publics pour apprendre les meilleures représentations possibles de ces structures, et de transférer cette connaissance globale vers des jeux de données plus restreints. Nous nous inspirons pour ce faire de méthodes utilisées en traitement automatique des langages naturels. Pour les mettre en œuvre, des travaux d'ordre plus théorique ont été nécessaires, notamment sur le problème d'isomorphisme de graphes. Les résultats obtenus sur des tâches de classification/régression sont au moins compétitifs avec l'état de l'art, voire meilleurs, en particulier sur des jeux de données restreints, attestant des possibilités d'apprentissage par transfert sur ce domaine.

International audience; Graphs increasingly stand out as an essential data structurein the field of data sciences. To study graphs, or sub-graphs, that char-acterize a set of observations, it is necessary to describe them formally,in order to characterize equivalence relations that make sense in thescope of the considered application domain. Hence we seek to define acanonical graph notation, so that two isomorphic (sub) graphs have thesame canonical form. Such notation could subsequently be used to indexand retrieve graphs or to embed them efficiently in some metric space.Sequential optimized algorithms solving this problem exist, but do notdeal with labeled edges, a situation that occurs in important applicationdomains such as chemistry. We present in this article a new algorithmbased on graph rewriting that provides a general and complete solution tothe graph canonization problem. Although not reported here, the formalproof of the validity of our algorithm has been established. This claim isclearly supported empirically by our experimentation on synthetic com-binatorics as well as natural graphs. Furthermore, our algorithm supportsdistributed implementations, leading to efficient computing perspectives.

A k-block in a graph G is a maximal set of at least k vertices no two of which can be separated in G by removing less than k vertices. It is separable if there exists a tree-decomposition of adhesion less than k of G in which this k-block appears as a part. Carmesin et al. proved that every finite graph has a canonical tree-decomposition of adhesion less than k that distinguishes all its k-blocks and tangles of order k. We construct such tree-decompositions with the additional property that every separable k-block is equal to the unique part in which it is contained. This proves a conjecture of Diestel.

We give a fixed-parameter tractable algorithm that, given a parameter $k$ and two graphs $G_1,G_2$, either concludes that one of these graphs has treewidth at least $k$, or determines whether $G_1$ and $G_2$ are isomorphic. The running time of the algorithm on an $n$-vertex graph is $2^{O(k^5\log k)}\cdot n^5$, and this is the first fixed-parameter algorithm for Graph Isomorphism parameterized by treewidth. Our algorithm in fact solves the more general canonization problem. We namely design a procedure working in $2^{O(k^5\log k)}\cdot n^5$ time that, for a given graph $G$ on $n$ vertices, either concludes that the treewidth of $G$ is at least $k$, or: * finds in an isomorphic-invariant way a graph $\mathfrak{c}(G)$ that is isomorphic to $G$; * finds an isomorphism-invariant construction term --- an algebraic expression that encodes $G$ together with a tree decomposition of $G$ of width $O(k^4)$. Hence, the isomorphism test reduces to verifying whether the computed isomorphic copies or the construction terms for $G_1$ and $G_2$ are equal., Full version of a paper presented at FOCS 2014

Discontinuity and polymorphism of a library function create two challenges for library function identification, which is a key technique in reverse engineering. A new hybrid representation of dependence graph and control flow graph called Execution Flow Graph (EFG) is introduced to describe the semantics of binary code. Library function identification turns to be a subgraph isomorphism testing problem since the EFG of a library function instance is isomorphic to the sub-EFG of this library function. Subgraph isomorphism detection is time-consuming. Thus, we introduce a new representation called Reduced Execution Flow Graph (REFG) based on EFG to speed up the isomorphism testing. We have proved that EFGs are subgraph isomorphic as long as their corresponding REFGs are subgraph isomorphic. The high efficiency of the REFG approach in subgraph isomorphism detection comes from fewer nodes and edges in REFGs and new lossless filters for excluding the unmatched subgraphs before detection. Experimental results show that precisions of both the EFG and REFG approaches are higher than the state-of-the-art tool and the REFG approach sharply decreases the processing time of the EFG approach with consistent precision and recall.

The maximum common subgraph isomorphism problem is a difficult graph problem, and the problem of finding the maximum common subgraph isomorphism problem is NP-hard. This means there is likely no algorithm that will be able to find the maximal isomorphic common subgraph in polynomial time because as the size of the graphs grows the search space for the solution will grow exponentially. This research provides a method that will approximate the maximum common subgraph isomorphism problem by producing a weighted graph, where the weights will give an indication of the probability that the associated link will be in the maximum common subgraph of the two input graphs. The experimental results show that the method can effectively generate a weighted graph containing most of the expected links that would exist in the maximum common subgraph of some given graphs with similar structures.

We study the complexity of isomorphism testing for boolean functions that are represented by decision trees or decision lists. Our results are the following:Isomorphism testing of rank 1 decision trees is complete for logspace.For any constant r ? 2 , isomorphism testing for rank r decision trees is polynomial-time equivalent to Graph Isomorphism. As a consequence of our reduction, we obtain our main result for decision trees: A 2 n ( log ? s ) O ( 1 ) time algorithm for isomorphism testing of decision trees of size s over n variables.The isomorphism problem for decision lists admits a Schaefer-type trichotomy: depending on the class of base functions, the isomorphism problem is either in L , or polynomial-time equivalent to Graph Isomorphism, or coNP -hard.

We give a new fpt algorithm testing isomorphism of $n$-vertex graphs of tree width $k$ in time $2^{k\operatorname{polylog} (k)}\operatorname{poly} (n)$, improving the fpt algorithm due to Lokshtanov, Pilipczuk, Pilipczuk, and Saurabh (FOCS 2014), which runs in time $2^{\mathcal{O}(k^5\log k)}\operatorname{poly} (n)$. Based on an improved version of the isomorphism-invariant graph decomposition technique introduced by Lokshtanov et al., we prove restrictions on the structure of the automorphism groups of graphs of tree width $k$. Our algorithm then makes heavy use of the group theoretic techniques introduced by Luks (JCSS 1982) in his isomorphism test for bounded degree graphs and Babai (STOC 2016) in his quasipolynomial isomorphism test. In fact, we even use Babai's algorithm as a black box in one place. We also give a second algorithm which, at the price of a slightly worse running time $2^{\mathcal{O}(k^2 \log k)}\operatorname{poly} (n)$, avoids the use of Babai's algorithm and, more importantly, has the additional benefit that it can also used as a canonization algorithm., 34 pages, 1 figures

In this paper, we introduce the streaming graph canonization problem. Its goal is finding a canonical representation of a sequence of graphs in a stream. Our model of a stream fixes the graph’s vertices and allows for fully dynamic edge changes, meaning it permits both addition and removal of edges. Our focus is on small graphs, since small graph isomorphism is an important primitive of many subgraph-based metrics, like motif analysis or frequent subgraph mining. We present an efficient data structure to approach this problem, namely a graph isomorphism discrete finite automaton and showcase its efficiency when compared to a non-streaming-aware method that simply recomputes the isomorphism information from scratch in each iteration.

In this paper, we study the parallel and the space complexity of the graph isomorphism problem (\(\mathsf {GI}\)) for several parameterizations.

The state-of-the-art tools for practical graph canonization are all based on the individualization-refinement paradigm, and their difference is primarily in the choice of heuristics they include and in the actual tool implementation. It is thus not possible to make a direct comparison of how individual algorithmic ideas affect the performance on different graph classes. We present an algorithmic software framework that facilitates implementation of heuristics as independent extensions to a common core algorithm. It therefore becomes easy to perform a detailed comparison of the performance and behavior of different algorithmic ideas. Implementations are provided of a range of algorithms for tree traversal, target cell selection, and node invariant, including choices from the literature and new variations. The framework readily supports extraction and visualization of detailed data from separate algorithm executions for subsequent analysis and development of new heuristics. Using collections of different graph classes, we investigate the effect of varying the selections of heuristics, often revealing exactly which individual algorithmic choice is responsible for particularly good or bad performance. On several benchmark collections, including a newly proposed class of difficult instances, we additionally find that our implementation performs better than the current state-of-the-art tools.

A regular graph is a type of graph that has particular features. Such graphs have great symmetry, which increases the difficulty of isomorphism identification. In this paper, we propose a new method to address regular graph isomorphism identification. We make some adjustments to the original circuit simulation (CS) method in the graph isomorphism identification process, using single-vertex excitation (SVE) in place of the original excitation, which could remove the confusion of corresponding vertices caused by the graph's symmetry. This new SVE method could widen the range of the application of the original CS method and create better performance in time consumption compared with other traditional methods. The performance of the SVE method is provided in this paper illuminating its practicality and superiority.

Let G be a finite group. A Cayley graph over G is a simple graph whose automorphism group has a regular subgroup isomorphic to G. A Cayley graph is called a CI-graph (Cayley isomorphism) if its isomorphic images are induced by automorphisms of G. A well-known result of Babai states that a Cayley graph Γ of G is a CI-graph if and only if all regular subgroups of Aut(Γ) isomorphic to G are conjugate in Aut(Γ). A semi-Cayley graph (also called bi-Cayley graph by some authors) over G is a simple graph whose automorphism group has a semiregular subgroup isomorphic to G with two orbits (of equal size). In this paper, we introduce the concept of SCI-graph (semi-Cayley isomorphism) and prove a Babai type theorem for semi-Cayley graphs. We prove that every semi-Cayley graph of a finite group G is an SCI-graph if and only if G is cyclic of order 3. Also, we study the isomorphism problem of a special class of semi-Cayley graphs.

A tropical graph $(H,c)$ consists of a graph $H$ and a (not necessarily proper) vertex-colouring $c$ of $H$. Given two tropical graphs $(G,c_1)$ and $(H,c)$, a homomorphism of $(G,c_1)$ to $(H,c)$ is a standard graph homomorphism of $G$ to $H$ that also preserves the vertex-colours. We initiate the study of the computational complexity of tropical graph homomorphism problems. We consider two settings. First, when the tropical graph $(H,c)$ is fixed; this is a problem called $(H,c)$-COLOURING. Second, when the colouring of $H$ is part of the input; the associated decision problem is called $H$-TROPICAL-COLOURING. Each $(H,c)$-COLOURING problem is a constraint satisfaction problem (CSP), and we show that a complexity dichotomy for the class of $(H,c)$-COLOURING problems holds if and only if the Feder-Vardi Dichotomy Conjecture for CSPs is true. This implies that $(H,c)$-COLOURING problems form a rich class of decision problems. On the other hand, we were successful in classifying the complexity of at least certain classes of $H$-TROPICAL-COLOURING problems., 27 pages, 13 figures, 1 table. Compared to the published version, this version includes all proofs and some additional figures

Graph-based data representations are an important research topic due to the suitability of this kind of data structure to model entities and the complex relations among them. In computer vision, graphs have been used to model images in order to add some high level information (relations) to the low-level representation of individual parts. How to deal with these representations for specific tasks is not easy due to the complexity of the data structure itself. In this paper we propose to use a graph mining technique for image classification, introducing approximate patterns discovery in the mining process in order to allow certain distortions in the data being modeled. We are proposing to combine a powerful graph-based image representation adapted to this specific task and frequent approximate subgraph (FAS) mining algorithms in order to classify images. In the case of image representation we are proposing to use more robust descriptors than our previous approach in this topic, and we also suggest a criterion to select the isomorphism threshold for the graph mining step. This proposal is tested in two well-known collections to show the improvement with respect to the previous related works. HighlightsWe propose a new framework for image classification, which uses frequent approximate subgraph patterns as features.We propose to compute automatically the substitution matrices needed in the process, instead of using expert knowledge.We propose to use a new graph-based image representation.We propose a criterion for selecting isomorphism threshold for the graph mining process.

Special issue PRIMA 2013, The graph isomorphism (GI) problem asks whether two given graphs are isomorphic or not. The GI problem is quite basic and simple, however, it\textquoterights time complexity is a long standing open problem. The GI problem is clearly in NP, no polynomial time algorithm is known, and the GI problem is not NP-complete unless the polynomial hierarchy collapses. In this paper, we survey the computational complexity of the problem on some graph classes that have geometric characterizations. Sometimes the GI problem becomes polynomial time solvable when we add some restrictions on some graph classes. The properties of these graph classes on the boundary indicate us the essence of difficulty of the GI problem. We also show that the GI problem is as hard as the problem on general graphs even for grid unit intersection graphs on a torus, that partially solves an open problem.

We study the problem of computing canonical forms for graphs and hypergraphs under Abelian group action and show tight complexity bounds. Our approach is algebraic. We transform the problem of computing canonical forms for graphs to the problem of computing canonical forms for associated algebraic structures, and we develop parallel algorithms for these associated problems. 1. In our first result we show that the problem of computing canonical labelings for hypergraphs of color class size 2 is logspace Turing equivalent to solving a system of linear equations over the field \(\mathbb {F} _{2}\). This implies a deterministic NC2 algorithm for the problem. 2. Similarly, we show that the problem of canonical labeling graphs and hypergraphs under arbitrary Abelian permutation group action is fairly well characterized by the problem of computing the integer determinant. In particular, this yields deterministic NC3 and randomized NC2 algorithms for the problem.

Given a pattern graph H with l edges, and a host graph G guaranteed to contain at most one occurrence of a subgraph isomorphic to H , we show that the time complexity of the problem of finding such an occurrence if any in G as well as that of the decision version of the problem are within a multiplicative factor O l of the time complexity for the corresponding problem in the general case, when G may contain several occurrences of H . It follows that for pattern graphs of constant size, the aforementioned uniqueness guarantee cannot yield any asymptotic speed up. We also derive analogous results with the analogous multiplicative factor linear in the number of vertices of H in the induced case when occurrences of induced subgraphs of G isomorphic to H are sought.

We show that every problem in the complexity class SZK (Statistical Zero Knowledge) is efficiently reducible to the Minimum Circuit Size Problem (MCSP). In particular Graph Isomorphism lies in RP MCSP. This is the first theorem relating the computational power of Graph Isomorphism and MCSP, despite the long history these problems share, as candidate NP-intermediate problems., 39th International Symposium, MFCS 2014, Budapest, Hungary, August 25-29, 2014

We study the complexity of determining whether a polytope given by its vertices or facets is combinatorially isomorphic to its polar dual. We prove that this problem is Graph Isomorphism hard, and that it is Graph Isomorphism complete if and only if Vertex Enumeration is Graph Isomorphism easy. To the best of our knowledge, this is the first problem that is not equivalent to Vertex Enumeration and whose complexity status has a non-trivial impact on the complexity of Vertex Enumeration irrespective of whether checking Self-duality turns out to be strictly harder than Graph Isomorphism or equivalent to Graph Isomorphism. The constructions employed in the proof yield a class of self-dual polytopes that are interesting on their own. In particular, this class of self-dual polytopes has the property that the facet-vertex incident matrix of the polytope is transposable if and only if the matrix is symmetrizable as well. As a consequence of this construction, we also prove that checking self-duality of a polytope, given by its facet-vertex incidence matrix, is Graph Isomorphism complete, thereby answering a question of Kaibel and Schwartz.

The partial order 〈E(R)∪{∅},⊂〉, where E(R) is the set of isomorphic subgraphs of the Rado graph R, is investigated. The order types of maximal chains in this poset are characterized as the order types of compact sets of reals having the minimum non-isolated.

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Querying graph data is a fundamental problem that witnesses an increasing interest especially for massive structured data where graphs come as a promising alternative to relational databases for big data modeling. However, querying graph data is different and more complex than querying relational table-based data. The main task involved in querying graph data is subgraph isomorphism search which is an NP-complete problem. Subgraph isomorphism search, is an important problem which is involved in various domains such as pattern recognition, social network analysis, biology, etc. It consists to enumerate the subgraphs of a data graph that match a query graph. The most known solutions of this problem are backtracking-based. They explore a large search space which results in a high computational cost when we deal with massive graph data. To reduce time and memory space complexity of subgraph isomorphism search. We propose to use compressed graphs. In our approach, subgraph isomorphism search is achieved on compressed representations of graphs without decompressing them. Graph compression is performed by grouping vertices into super vertices. This concept is known, in graph theory, as modular decomposition. It is used to generate a tree representation of a graph that highlights groups of vertices that have the same neighbors. With this compression we obtain a substantial reduction of the search space and consequently a significant saving in the processing time. We also propose a novel encoding of vertices that simplifies the filtering of the search space. This new mechanism is called compact neighborhood Index (CNI). A CNI distills all the information around a vertex in a single integer. This simple neighborhood encoding reduces the time complexity of vertex filtering from cubic to quadratic which is considerable for big graphs. We propose also an iterative local global filtering algorithm that relies on the characteristics of CNIs to ensure a global pruning of the search space.We evaluated our approaches on several real-word datasets and compared them with the state of the art algorithms

We investigate the computational complexity of the following restricted variant of Subgraph Isomorphism : given a pair of connected graphs G = ( V G , E G ) and H = ( V H , E H ) , determine if H is isomorphic to a spanning subgraph of G . The problem is NP-complete in general, and thus we consider cases where G and H belong to the same graph class such as the class of proper interval graphs, of trivially perfect graphs, and of bipartite permutation graphs. For these graph classes, several restricted versions of Subgraph Isomorphism such as Hamiltonian Path , Clique , Bandwidth , and Graph Isomorphism can be solved in polynomial time, while these problems are hard in general.

The search for an easily computable, finite, complete set of graph invariants remains a challenging research topic. All measures characterizing the topology of a graph that have been developed thus far exhibit some degree of degeneracy, i.e., an inability to distinguish between non-isomorphic graphs. In this paper, we show that certain graph invariants can be useful in substantially reducing the computational complexity of isomorphism testing. Our findings are underpinned by numerical results based on a large scale statistical analysis.

Data mining in structured and semi-structured data focuses on frequent data values. However, in graph data mining, the focus is on common specific topologies. Graph mining, although its ubiquity, is a difficult task since it requires subgraph isomorphism which is known to be NP-complete. In order to effectively prune the search space and thereby save computational time, a graph mining algorithm requires that the support measure of a pattern to be no greater than that of its subpatterns. This property of the support measure is referred to in the literature as the down-closure, anti-monotonicity or admissibility. Unfortunately, when mining a single labeled graph, simply counting the occurrences of a graph pattern may not have the down-closure property. For this, most existing approaches mine frequent substructures in a set of labeled graphs (called also the transactional setting) and few efforts have been devoted to mining frequent globally distributed substructures in a single labeled graph. In this paper, we propose a graph mining algorithm, called NODAR(Non-Overlapping embeDding based grAph mineR), for computing common and globally distributed substructures in a single labeled graph. NODAR adopts the Depth-First Search (DFS) strategy and is based on the SMNOES (Size of Maximum Non Overlapping Embedding Set) as support measure. The core idea of NODAR is to automatically extract frequent subpatterns; and thus without frequency computation thanks to the down-closure property of SMNOES. By adopting this strategy in the computation of frequent substructures, NODAR reduces the number of subgraph isomorphism tests needed to compute pattern frequencies. Experimental results on monograph and transactional graph databases; and comparison with well-known probabilistic and exact algorithms; prove the efficacy of NODAR.