Graph Isomorphism Problem Research Articles

On Borel complexity of the isomorphism problems for graph related classes of Lie algebras and finite p-groups

We reduce the isomorphism problem for undirected graphs without loops to the isomorphism problems for some class of finite-dimensional 2-step nilpotent Lie algebras over a field and for some class of finite p-groups. We show that the isomorphism problem for graphs is harder than the two latter isomorphism problems in the sense of Borel reducibility. A computable analogue of Borel reducibility was introduced by S. Coskey, J. D. Hamkins, and R. Miller (2012). A relation of the isomorphism problem for undirected graphs to the well-known problem of classifying pairs of matrices over a field (up to similarity) is also studied.

On spectral properties for graph matching and graph isomorphism problems

Problems related to graph matching and isomorphisms are very important both from a theoretical and practical perspective, with applications ranging from image and video analysis to biological and biomedical problems. The graph matching problem is challenging from a computational point of view, and therefore different relaxations are commonly used. Although common relaxations techniques tend to work well for matching perfectly isomorphic graphs, it is not yet fully understood under which conditions the relaxed problem is guaranteed to obtain the correct answer. In this paper, we prove that the graph matching problem and its most common convex relaxation, where the matching domain of permutation matrices is substituted with its convex hull of doubly-stochastic matrices, are equivalent for a certain class of graphs, such equivalence being based on spectral properties of the corresponding adjacency matrices. We also derive results about the automorphism group of a graph, and provide fundamental spectral properties of the adjacency matrix.

Graph Isomorphism Completeness for Trapezoid Graphs

The complexity of the graph isomorphism problem for trapezoid graphs has been open over a decade. This paper shows that the problem is GI-complete. More precisely, we show that the graph isomorphism problem is GI-complete for comparability graphs of partially ordered sets with interval dimension 2 and height 3. In contrast, the problem is known to be solvable in polynomial time for comparability graphs of partially ordered sets with interval dimension at most 2 and height at most 2.

Detecting and Counting Small Pattern Graphs

We study the induced subgraph isomorphism problem and the general subgraph isomorphism problem for small pattern graphs. We present a new general method for detecting induced subgraphs of a host graph isomorphic to a fixed pattern graph by reduction to polynomial testing for nonidentity with zero over a field of finite characteristic. It yields new upper time bounds for several pattern graphs on five vertices and provides an alternative combinatorial method for the majority of pattern graphs on four and three vertices. Since our method avoids the large overhead of fast matrix multiplication, it can be of practical interest even for larger pattern graphs. Next, we derive new upper time bounds on counting the number of isomorphisms between a fixed pattern graph with an independent set of size $s$ and a subgraph of the host graph. We also consider a weighted version of the counting problem, when one counts the number of isomorphisms between the pattern graph and lightest subgraphs, providing a slightly slowe...

On the generic complexity of the searching graph isomorphism problem

AbstractA generic-case approach to algorithmic problems was suggested by Myasnikov, Kapovich, Schupp and Shpilrain in 2003. This approach studies the behavior of an algorithm on typical inputs and ignores the rest of the inputs. In this paper we consider generic complexity of the searching graph isomorphism problem. We fit this problem in the frameworks of generic complexity and prove that its natural subproblem is generically hard provided that the searching graph isomorphism problem is hard in the worst case.

Graph isomorphism for graph classes characterized by two forbidden induced subgraphs

We study the complexity of the Graph Isomorphism problem on graph classes that are characterized by a finite number of forbidden induced subgraphs, focusing mostly on the case of two forbidden subgraphs. We show hardness results and develop techniques for the structural analysis of such graph classes. Applied to the case of two forbidden subgraphs we obtain the following main result: A dichotomy into isomorphism complete and polynomial-time solvable graph classes for all but finitely many cases, whenever neither forbidden graph is a clique, a pan, or a complement of these graphs. Further reducing the remaining open cases we show that (with respect to graph isomorphism) forbidding a pan is equivalent to forbidding a clique of size three.

The graph isomorphism problem on geometric graphs

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.

On the automorphism groups of strongly regular graphs II

We derive strong constraints on the automorphism groups of strongly regular (SR) graphs, resolving old problems motivated by Peter Cameron's 1981 description of large primitive groups.Trivial SR graphs are the disjoint unions of cliques of equal size and their complements. Graphic SR graphs are the line-graphs of cliques and of regular bipartite cliques (complete bipartite graphs with equal parts) and their complements.We conjecture that the order of the automorphism group of a non-trivial, non-graphic SR graph is quasi-polynomially bounded, i.e., it is at most exp⁡((log⁡n)C) for some constant C, where n is the number of vertices.While the conjecture remains open, we find surprisingly strong bounds on important parameters of the automorphism group. In particular, we show that the order of every automorphism is O(n8), and in fact O(n) if we exclude the line-graphs of certain geometries. We prove the conjecture for the case when the automorphism group is primitive; in this case we obtain a nearly tight n1+log2⁡n bound.We obtain these bounds by bounding the fixicity of the automorphism group, i.e., the maximum number of fixed points of non-identity automorphisms, in terms of the second largest (in magnitude) eigenvalue and the maximum number of pairwise common neighbors of a regular graph. We connect the order of the automorphisms to the fixicity through an old lemma by Ákos Seress and the author.We propose to extend these investigations to primitive coherent configurations and offer problems and conjectures in this direction. Part of the motivation comes from the complexity of the Graph Isomorphism problem.

A quantum-walk-inspired adiabatic algorithm for solving graph isomorphism problems

We present a quantum algorithm for solving graph isomorphism problems that is based on an adiabatic protocol. We use a collection of continuous time quantum walks, each one generated by an XY Hamiltonian, to visit the configuration space. In this way we avoid a diffusion over all the possible configurations and significantly reduce the dimensionality of the accessible Hilbert space. Within this restricted space, the graph isomorphism problem can be translated into searching for a satisfying assignment to a 2-SAT (satisfiable) formula and mapped onto a 2-local Hamiltonian without resorting to perturbation gadgets or projective techniques. We present an analysis of the time for execution of the algorithm on small graph isomorphism problem instances and discuss the issue of an implementation of the proposed adiabatic scheme on current quantum computing hardware.

Zero-Divisor Graphs for Group Rings

Let R be a commutative ring with 1 ≠ 0, G be a nontrivial finite group, and let Z(R) be the set of zero divisors of R. The zero-divisor graph of R is defined as the graph Γ(R) whose vertex set is Z(R)* = Z(R)∖{0} and two distinct vertices a and b are adjacent if and only if ab = 0. In this paper, we investigate the interplay between the ring-theoretic properties of group rings RG and the graph-theoretic properties of Γ(RG). We characterize finite commutative group rings RG for which either diam(Γ(RG)) ≤2 or gr(Γ(RG)) ≥4. Also, we investigate the isomorphism problem for zero-divisor graphs of group rings. First, we show that the rank and the cardinality of a finite abelian p-group are determined by the zero-divisor graph of its modular group ring. With the notion of zero-divisor graphs extended to noncommutative rings, it is also shown that two finite semisimple group rings are isomorphic if and only if their zero-divisor graphs are isomorphic. Finally, we show that finite noncommutative reversible group rings are determined by their zero-divisor graphs.

Graph isomorphism and adiabatic quantum computing

In the Graph Isomorphism problem two N-vertex graphs G and G' are given and the task is to determine whether there exists a permutation of the vertices of G that preserves adjacency and transforms G into G'. If yes, then G and G' are said to be isomorphic; otherwise they are non-isomorphic. The GI problem is an important problem in computer science and is thought to be of comparable difficulty to integer factorization. In this paper we present a quantum algorithm that solves arbitrary instances of GI and can also determine all automorphisms of a given graph. We show how the GI problem can be converted to a combinatorial optimization problem that can be solved using adiabatic quantum evolution. We numerically simulate the algorithm's quantum dynamics and show that it correctly: (i) distinguishes non-isomorphic graphs; (ii) recognizes isomorphic graphs; and (iii) finds all automorphisms of a given graph G. We then discuss the GI quantum algorithm's experimental implementation, and close by showing how it can be leveraged to give a quantum algorithm that solves arbitrary instances of the NP-Complete Sub-Graph Isomorphism problem.

A Practical Graph Isomorphism Algorithm with Vertex Canonical Labeling

The vertex canonical labeling technique is one of the powerful methods in solving the graph isomorphism problem. However, some famous algorithms relied upon this technique are incomplete due to their non-zero probability of rejection. In this paper, we advance a new method of vertex canonical labeling and propose a complete graph isomorphism algorithm with depth-first backtracking. The time complexity of this algorithm is O( n ^ a ) where n ^ a is the number of backtracking points and ( h -1)/2≤ a ≤( h +1)/2 for h =log n in most cases. Finally, the proposed algorithm is compared with other researches on many types of graphs. The performance results validated that our algorithm is efficient for a wide variety of families of graphs.

The comparison of tree-sibling time consistent phylogenetic networks is graph isomorphism-complete.

Several polynomial time computable metrics on the class of semibinary tree-sibling time consistent phylogenetic networks are available in the literature; in particular, the problem of deciding if two networks of this kind are isomorphic is in P. In this paper, we show that if we remove the semibinarity condition, then the problem becomes much harder. More precisely, we prove that the isomorphism problem for generic tree-sibling time consistent phylogenetic networks is polynomially equivalent to the graph isomorphism problem. Since the latter is believed not to belong to P, the chances are that it is impossible to define a metric on the class of all tree-sibling time consistent phylogenetic networks that can be computed in polynomial time.

Efficient construction of split-plot design catalogs using graphs

Fractional-factorial split-plot designs are useful variants of the traditional fractional-factorial designs. They incorporate practical constraints on the randomization of experiment runs. Catalogs of split-plot designs are useful to practitioners as they provide a means of selecting the best design suitable for their task. However, the construction of these catalogs is computationally challenging as it requires comparing designs for isomorphism, usually in a large collection. This article presents an efficient approach for constructing these catalogs by transforming the design isomorphism problem to a graph isomorphism problem. A new graph representation of split-plot designs is presented to achieve this aim. Using examples it is shown how these graph representations can be extended to certain other classes of factorial designs for solving the (corresponding) design isomorphism problem. The efficacy of this approach is demonstrated by presenting catalogs of two-level regular fractional factorial split-plot designs of up to 4096 runs, which is much larger than available in existing literature.

Practical graph isomorphism, II

We report the current state of the graph isomorphism problem from the practical point of view. After describing the general principles of the refinement-individualization paradigm and pro ving its validity, we explain how it is implemented in several of the key implementations. In particular, we bring the description of the best known program nauty up to date and describe an innovative approach called Traces that outperforms the competitors for many difficult graph classes. Detailed comparisons against saucy, Bliss and conauto are presented.

QAPV: a polynomial invariant for graph isomorphism testing

To each instance of the Quadratic Assignment Problem (QAP) a relaxed instance can be associated. Both variances of their solution values can be calculated in polynomial time. The graph isomorphism problem (GIP) can be modeled as a QAP, associating its pair of data matrices with a pair of graphs of the same order and size. We look for invariant edge weight functions for the graphs composing the instances in order to try to find quantitative differences between variances that could be associated with the absence of isomorphism. This technique is sensitive enough to show the effect of a single edge exchange between two regular graphs of up to 3,000 vertices and 300,000 edges with degrees up to 200. Planar graph pairs from a dense family up to 300,000 vertices were also discriminated. We conjecture the existence of functions able to discriminate non-isomorphic pairs for every instance of the problem.

The Graph Isomorphism Problem and approximate categories

It is unknown whether two graphs can be tested for isomorphism in polynomial time. A classical approach to the Graph Isomorphism Problem is the d-dimensional Weisfeiler–Lehman algorithm. The d-dimensional WL-algorithm can distinguish many pairs of graphs, but the pairs of non-isomorphic graphs constructed by Cai, Fürer and Immerman it cannot distinguish. If d is fixed, then the WL-algorithm runs in polynomial time. We will formulate the Graph Isomorphism Problem as an Orbit Problem: Given a representation V of an algebraic group G and two elements v1,v2∈V, decide whether v1 and v2 lie in the same G-orbit. Then we attack the Orbit Problem by constructing certain approximate categories Cd, d∈N={1,2,3,…} whose objects include the elements of V. We show that v1 and v2 are not in the same orbit by showing that they are not isomorphic in the category Cd for some d∈N. For every d this gives us an algorithm for isomorphism testing. We will show that the WL-algorithms reduce to our algorithms, but that our algorithms cannot be reduced to the WL-algorithms. Unlike the Weisfeiler–Lehman algorithm, our algorithm can distinguish the Cai–Fürer–Immerman graphs in polynomial time.

An Optimized Circuit Simulation Method for the Identification of Isomorphic Disconnected Graphs

Many algorithms exist to deal with the problem of graph isomorphism, but most of the contemporary algorithms like VF2 have a bad performance for identifying isomorphism of disconnected graphs. In this paper, we propose a method that can filter and remove single vertices so as to eliminate the dependence of the relationship between all vertices which causes the loss of effectiveness. Tests on disconnected graphs demonstrate that this method can solve the problem of identifying isomorphic disconnected graphs in polynomial time.

A Method of Colored Graph Isomorphism Problem

A Method of Colored Graph Isomorphism Problem

Fast-On*: An Extended Algorithm for Graph Isomorphism Problem and Graph Query Processing

Fast-On*: An Extended Algorithm for Graph Isomorphism Problem and Graph Query Processing