Graph Isomorphism Research Articles

Isomorphism, canonization, and definability for graphs of bounded rank width

We investigate the interplay between the graph isomorphism problem, logical definability, and structural graph theory on a rich family of dense graph classes: graph classes of bounded rank width. We prove that the combinatorial Weisfeiler-Leman algorithm of dimension (3 k + 4) is a complete isomorphism test for the class of all graphs of rank width at most k. A consequence of our result is the first polynomial time canonization algorithm for graphs of bounded rank width. Our second main result addresses an open problem in descriptive complexity theory: we show that fixed-point logic with counting expresses precisely the polynomial time properties of graphs of bounded rank width.

Technical perspective: A logical step toward the graph isomorphism problem

Technical perspective: A logical step toward the graph isomorphism problem

Logical complexity of induced subgraph isomorphism for certain families of graphs

We investigate the problem of the most efficient first-order definition of the property of containing an induced subgraph isomorphic to a given pattern graph, which is closely related to the time complexity of the decision problem for this property. We derive a series of new bounds for the minimum quantifier depth of a formula defining this property for pattern graphs on five vertices, as well as for disjoint unions of isomorphic complete multipartite graphs. Moreover, we prove that for any there exists a graph on vertices and a first-order formula of quantifier depth at most that defines the property of containing an induced subgraph isomorphic to this graph. Bibliography: 12 titles.

Quotients of the Gordian and H(2)-Gordian graphs

The Gordian graph and H(2)-Gordian graphs of knots are abstract graphs whose vertex sets represent isotopy classes of unoriented knots, and whose edge sets record whether pairs of knots are related by crossing changes or H(2)-moves, respectively. We investigate quotients of these graphs under equivalence relations defined by several knot invariants including the determinant, the span of the Jones polynomial, and an invariant related to tricolorability. We show, in all cases considered, that the quotient graphs are Gromov hyperbolic. We then prove a collection of results about the graph isomorphism type of the quotient graphs. In particular, we find that the H(2)-Gordian graph of links modulo the relation induced by the span of the Jones polynomial is isomorphic with the complete graph on infinitely many vertices.

Scalable generalized median graph estimation and its manifold use in bioinformatics, clustering, classification, and indexing

In this paper, we present GMG-BCU—a local search algorithm based on block coordinate update for estimating a generalized median graph for a given collection of labeled or unlabeled input graphs. Unlike all competitors, GMG-BCU is designed for both discrete and continuous label spaces and can be configured to run in linear time w.r.t. the size of the graph collection whenever median node and edge labels are computable in linear time. These properties make GMG-BCU usable for applications such as differential microbiome data analysis, graph classification, clustering, and indexing. We also prove theoretical properties of generalized median graphs, namely, that they exist under reasonable assumptions which are met in almost all application scenarios, that they are in general non-unique, that they are NP-hard to compute and APX-hard to approximate, and that no polynomial α-approximation exists for any α unless the graph isomorphism problem is in P. Extensive experiments on six different datasets show that our heuristic GMG-BCU always outperforms the state of the art in terms of runtime or quality (on most datasets, both w.r.t. runtime and quality), that it is the only available heuristic which can cope with collections containing several thousands of graphs, and that it shows very promising potential when used for the aforementioned applications. GMG-BCU is freely available on GitHub: https://github.com/dbblumenthal/gedlib/.

Computational Complexity of Computing Symmetries in Finite-Domain Planning

Symmetry-based pruning is a powerful method for reducing the search effort in finitedomain planning. This method is based on exploiting an automorphism group connected to the ground description of the planning task { these automorphisms are known as structural symmetries. In particular, we are interested in the StructSym problem where the generators of this group are to be computed. It has been observed in practice that the StructSym problem is surprisingly easy to solve. We explain this phenomenon by showing that StructSym is GI-complete, i.e., the graph isomorphism problem is polynomial-time equivalent to it and, consequently, solvable in quasi-polynomial time. This implies that it is solvable substantially faster than most computationally hard problems encountered in AI. We accompany this result by identifying natural restrictions of the planning task and its causal graph that ensure that StructSym can be solved in polynomial time. Given that the StructSym problem is GI-complete and thus solvable quite efficiently, it is interesting to analyse if other symmetries (than those that are encompassed by the StructSym problem) can be computed and/or analysed efficiently, too. To this end, we present a highly negative result: checking whether there exists an automorphism of the state transition graph that maps one state s into another state t is a PSPACE-hard problem and, consequently, at least as hard as the planning problem itself.

Searching for square-complementary graphs: Complexity of recognition and further nonexistence results

A graph is square-complementary (squco, for short) if its square and complement are isomorphic. We prove that there are no squco graphs with girth 6, that every bipartite graph is an induced subgraph of a bipartite squco graph, that the problem of recognizing squco graphs is graph isomorphism complete, and that no nontrivial squco graph is both bipartite and planar. These results resolve three of the open problems posed by Milanič, Pedersen, Pellicer, and Verret in 2014.

Neighborhood degree sum-based molecular descriptors of fractal and Cayley tree dendrimers.

Topological index is a connection between the chemical structure and the real number that remains invariant under graph isomorphism. In structure–property and structure–activity modeling, topological indices are considered as essential molecular descriptors to predict different physicochemical properties of molecule. Dendrimers are considered to be the most significant, commercially accessible basic components in nanotechnology. In this report, some neighborhood degree sum-based molecular descriptors are obtained for the fractal tree and the Cayley tree dendrimers. Neighborhood M-polynomial yields a family of topological indices for a molecular graph in less time compared to the usual computation from their definitions. Some indices are obtained using neighborhood M-polynomial approach. In addition, some multiplicative neighborhood degree sum-based molecular descriptors are evaluated for fractal and Cayley tree dendrimers. The graphical representations of the outcomes are presented. A comparative study of the findings with some well-known degree-based indices is performed. Usefulness of the descriptors in modeling different properties and activities is discussed.

Synchronous linear constraint system games

Mathematical models of quantum mechanics can be studied and distinguished using nonlocal games. We discuss a class of nonlocal games called synchronous linear constraint system (syncLCS) games. We unify two algebraic approaches to studying syncLCS games and relate these games to nonlocal games played on graphs, known as graph isomorphism games. In more detail, syncLCS games are nonlocal games that verify whether or not two players share a solution to a given system of equations. Two algebraic objects associated with these games encode information about the existence of perfect strategies. They are called the game algebra and the solution group. Here, we show that these objects are essentially the same, i.e., the game algebra is a suitable quotient of the group algebra of the solution group. We also demonstrate that syncLCS games are equivalent to graph isomorphism games on a pair of graphs parameterized by the linear system.

Implementing graph-theoretic quantum algorithms on a silicon photonic quantum walk processor.

Applications of quantum walks can depend on the number, exchange symmetry and indistinguishability of the particles involved, and the underlying graph structures where they move. Here, we show that silicon photonics, by exploiting an entanglement-driven scheme, can realize quantum walks with full control over all these properties in one device. The device we realize implements entangled two-photon quantum walks on any five-vertex graph, with continuously tunable particle exchange symmetry and indistinguishability. We show how this simulates single-particle walks on larger graphs, with size and geometry controlled by tuning the properties of the composite quantum walkers. We apply the device to quantum walk algorithms for searching vertices in graphs and testing for graph isomorphisms. In doing so, we implement up to 100 sampled time steps of quantum walk evolution on each of 292 different graphs. This opens the way to large-scale, programmable quantum walk processors for classically intractable applications.

Combinatorial entanglement

We present new combinatorial objects, which we call grid-labelled graphs, and show how these can be used to represent the quantum states arising in a physical scenario which we refer to as the faulty emitter scenario: we have a machine designed to emit a particular quantum state on demand, but which can make an error and emit a different one. The device is able to produce a list of candidate states which can be used as a kind of debugging information for testing if the resulting state is entangled. By reformulating the Peres-Horodecki and matrix realignment entanglement criteria we are able to capture some characteristic features of entanglement: we construct new bound entangled states, and demonstrate the limitations of matrix realignment. We show how the notion of local operations and classical communication (LOCC) is related to a generalisation of the graph isomorphism problem. We give a simple proof that asymptotically almost surely, grid-labelled graphs associated to very sparse density matrices are entangled. We develop tools for enumerating grid-labelled graphs that satisfy the Peres-Horodecki criterion up to a fixed number of vertices, and propose various computational problems for these objects, the computational complexity of which we leave as an open problem. The proposed mathematical framework also suggests new combinatorial and algebraic ways for describing the structure of graphs.

Direct product primality testing of graphs is GI-hard

We investigate the computational complexity of the graph primality testing problem with respect to the direct product (also known as Kronecker, cardinal or tensor product). In [1] Imrich proves that both primality testing and a unique prime factorization can be determined in polynomial time for (finite) connected and nonbipartite graphs. The author states as an open problem how results on the direct product of nonbipartite, connected graphs extend to bipartite connected graphs and to disconnected ones. In this paper we partially answer this question by proving that the graph isomorphism problem is polynomial-time many-one reducible to the graph compositeness testing problem (the complement of the graph primality testing problem). As a consequence of this result, we prove that the graph isomorphism problem is polynomial-time Turing reducible to the primality testing problem. Our results show that connectedness plays a crucial role in determining the computational complexity of the graph primality testing problem.

Graph isomorphism restricted by lists

The complexity of graph isomorphism (GraphIso ) is a famous problem in computer science. For graphs G and H, it asks whether they are the same up to a relabeling of vertices. In 1981, Lubiw proved that list restricted graph isomorphism (ListIso ) is NP -complete: for each u∈V(G), we are given a list L(u)⊆V(H) of possible images of u. After 35 years, we revive the study of this problem and consider which results for GraphIso can be modified to solve ListIso. We prove: 1) Under certain conditions, GI -completeness of a class of graphs implies NP -completeness of ListIso. 2) Several combinatorial algorithms for GraphIso can be modified to solve ListIso: for trees, planar graphs, interval graphs, circle graphs, permutation graphs, and bounded treewidth graphs. 3) ListIso is NP -complete for cubic colored graphs with sizes of color classes bounded by 8 with all lists of size at most 3.

Bisynchronous Games and Factorizable Maps

We introduce a new class of non-local games and corresponding densities, which we call bisynchronous. Bisynchronous games are a subclass of synchronous games and exhibit many interesting symmetries when the algebra of the game is considered. We develop a close connection between these non-local games and the theory of quantum groups which recently surfaced in studies of graph isomorphism games. When the number of inputs is equal to the number of outputs, we prove that a bisynchronous density arises from a trace on the quantum permutation group. Each bisynchronous density gives rise to a completely positive map, and we prove that these maps are factorizable maps.

Graphs with Two Moplexes

Moplexes are natural graph structures that arise when lifting Dirac’s classical theorem from chordal graphs to general graphs. The notion is known to be closely related to lexicographic searches in graphs as well as to asteroidal triples, and has been applied in several algorithms related to graph classes such as interval graphs, claw-free, and diamond-free graphs. However, while every non-complete graph has at least two moplexes, little is known about structural properties of graphs with a bounded number of moplexes. The study of these graphs is, among others, motivated by the parallel between moplexes in general graphs and simplicial modules in chordal graphs: unlike in the moplex setting, properties of chordal graphs with a bounded number of simplicial modules are well understood. For instance, chordal graphs having at most two simplicial modules are interval.In this work we initiate an investigation of k-moplex graphs, which are defined as graphs containing at most k moplexes. Of particular interest is the smallest nontrivial case, k = 2, which forms a counterpart to the class of interval graphs. As our main structural result, we show that the class of connected 2-moplex graphs is sandwiched between the classes of proper interval graphs and cocomparability graphs; moreover, both inclusions are tight for hereditary classes. From a complexity theoretic viewpoint, this leads to the natural question of whether the presence of at most two moplexes guarantees a sufficient amount of structure to efficiently solve problems that are known to be intractable on cocomparability graphs, but not on proper interval graphs. We develop new reductions that answer this question negatively for two prominent problems fitting this profile, namely Graph Isomorphism and Max-Cut. Furthermore, for graphs with a higher number of moplexes, we lift the previously known result that graphs without asteroidal triples have at most two moplexes to the more general setting of larger asteroidal sets. We also discuss sufficient conditions for the existence of Hamiltonian paths in 2-moplex graphs as well as connections with avoidable vertices.

Graph isomorphism and Gaussian boson sampling

AbstractWe introduce a connection between a near-term quantum computing device, specifically a Gaussian boson sampler, and the graph isomorphism problem. We propose a scheme where graphs are encoded into quantum states of light, whose properties are then probed with photon-number-resolving detectors. We prove that the probabilities of different photon-detection events in this setup can be combined to give a complete set of graph invariants. Two graphs are isomorphic if and only if their detection probabilities are equivalent. We present additional ways that the measurement probabilities can be combined or coarse-grained to make experimental tests more amenable. We benchmark these methods with numerical simulations on the Titan supercomputer for several graph families: pairs of isospectral nonisomorphic graphs, isospectral regular graphs, and strongly regular graphs.

Determining the Number of Disconnected Vertices Labeled Graphs of Order Six with the Maximum Number Twenty Parallel Edges and Containing No Loops

If there exist two vertices on a given graph that are not connected by a path, then we call that graph is disconnected. Given a graph with n vertices and m edges, then a lot of graphs can be constructed. In this paper, we discuss the number of disconnected vertices labeled graphs of order six (n = 6) with the maximum number of parallel edges is twenty. Moreover, a maximum number of edges that connect different pair of vertices is ten (parallel edges are counted as one) and containing no loops (isomorphic graphs are counted as one).

An Efficient Solution to Detect Common Topologies in Money Launderings Based on Coupling and Connection

In recent years, money laundering has become much easier to be achieved but more challenging to be detected than before, which has enormous adversary effects on finance, military, and other related fields. In the real-time scenario, every money laundering case has a unique structure in terms of transactions. It is not sufficient to detect suspicious behavior by just following the probability theory, where usually the thresholds are given by experts. Since the crime of money laundering is more prevalent and sophisticated nowadays, it will increase the complexity of the detection if the accounts with the personal information are combined with the form of the transaction topology. Hence, the graph topology analysis could be used for antimoney laundering tools. This article proposes eight common topologies based on coupling and connection from simple to much more complicated structures to solve various kinds of problems concerning money laundering in the real world. Moreover, we also propose an efficient solution based on graph and subgraph isomorphism and distance measurement to detect money laundering behavior. In this way, the detection of money laundering behavior will be more efficient and effective for various situations while referencing the proposed eight topological structures.

An upper bound of a generalized upper Hamiltonian number of a graph

In this article we study graphs with ordering of vertices, we define a generalization called a pseudoordering, and for a graph $H$ we define the $H$-Hamiltonian number of a graph $G$. We will show that this concept is a generalization of both the Hamiltonian number and the traceable number. We will prove equivalent characteristics of an isomorphism of graphs $G$ and $H$ using $H$-Hamiltonian number of $G$. Furthermore, we will show that for a fixed number of vertices, each path has a maximal upper $H$-Hamiltonian number, which is a generalization of the same claim for upper Hamiltonian numbers and upper traceable numbers. Finally we will show that for every connected graph $H$ only paths have maximal $H$-Hamiltonian number.

Permutations with restricted movement

A restricted permutation of a locally finite directed graph $ G = (V, E) $ is a vertex permutation $ \pi: V\to V $ for which $ (v, \pi(v))\in E $, for any vertex $ v\in V $. The set of such permutations, denoted by $ \Omega(G) $, with a group action induced from a subset of graph isomorphisms form a topological dynamical system. We focus on the particular case presented by Schmidt and Strasser [18] of restricted $ {\mathbb Z}^d $ permutations, in which $ \Omega(G) $ is a subshift of finite type. We show a correspondence between restricted permutations and perfect matchings (also known as dimer coverings). We use this correspondence in order to investigate and compute the topological entropy in a class of cases of restricted $ {\mathbb Z}^d $-permutations. We discuss the global and local admissibility of patterns, in the context of restricted $ {\mathbb Z}^d $-permutations. Finally, we review the related models of injective and surjective restricted functions.