Linear matching-time algorithm for the directed graph isomorphism problem

The graph isomorphism problem (GI) plays a central role in the theory of computational complexity and has importance in physics and chemistry as well \cite{kobler93,fortin96}. No polynomial-time algorithm for solving GI is known. We investigate classical and quantum physics-based polynomial-time algorithms for solving the graph isomorphism problem in which the graph structure is reflected in the behavior of a dynamical system. We show that a classical dynamical algorithm proposed by Gudkov and Nussinov \cite{gudkov02} as well as its simplest quantum generalization fail to distinguish pairs of non-isomorphic strongly regular graphs. However, by combining the algorithm of Gudkov and Nussinov with a construction proposed by Rudolph \cite{rudolph02} in which one examines a graph describing the dynamics of two particles on the original graph, we find an algorithm that successfully distinguishes all pairs of non-isomorphic strongly regular graphs that we tested with up to 29 vertices.

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Oriented Coloring on Recursively Defined Digraphs

Algorithms ◽

10.3390/a12040087 ◽

2019 ◽

Vol 12 (4) ◽

pp. 87 ◽

Cited By ~ 5

Author(s):

Frank Gurski ◽

Dominique Komander ◽

Carolin Rehs

Keyword(s):

Chromatic Number ◽

Linear Time ◽

Directed Graphs ◽

Graph Isomorphism ◽

Oriented Graph ◽

Time Algorithm ◽

Isomorphism Problem ◽

Np Hard ◽

Graph Isomorphism Problem ◽

Acyclic Digraph

Coloring is one of the most famous problems in graph theory. The coloring problem on undirected graphs has been well studied, whereas there are very few results for coloring problems on directed graphs. An oriented k-coloring of an oriented graph G = ( V , A ) is a partition of the vertex set V into k independent sets such that all the arcs linking two of these subsets have the same direction. The oriented chromatic number of an oriented graph G is the smallest k such that G allows an oriented k-coloring. Deciding whether an acyclic digraph allows an oriented 4-coloring is NP-hard. It follows that finding the chromatic number of an oriented graph is an NP-hard problem, too. This motivates to consider the problem on oriented co-graphs. After giving several characterizations for this graph class, we show a linear time algorithm which computes an optimal oriented coloring for an oriented co-graph. We further prove how the oriented chromatic number can be computed for the disjoint union and order composition from the oriented chromatic number of the involved oriented co-graphs. It turns out that within oriented co-graphs the oriented chromatic number is equal to the length of a longest oriented path plus one. We also show that the graph isomorphism problem on oriented co-graphs can be solved in linear time.

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The graph isomorphism problem on geometric graphs

Discrete Mathematics & Theoretical Computer Science ◽

10.46298/dmtcs.2076 ◽

2014 ◽

Vol Vol. 16 no. 2 (PRIMA 2013) ◽

Author(s):

Ryuhei Uehara

Keyword(s):

Polynomial Time ◽

Open Problem ◽

Graph Isomorphism ◽

Polynomial Time Algorithm ◽

Time Algorithm ◽

Isomorphism Problem ◽

Polynomial Hierarchy ◽

Intersection Graphs ◽

Graph Isomorphism Problem ◽

Graph Classes

Special issue PRIMA 2013 International audience 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.

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SPECTRUM BASED TECHNIQUES FOR GRAPH ISOMORPHISM

International Journal of Foundations of Computer Science ◽

10.1142/s0129054109006693 ◽

2009 ◽

Vol 20 (03) ◽

pp. 479-499

Author(s):

SANGUTHEVAR RAJASEKARAN ◽

VAMSI KUNDETI

Keyword(s):

Polynomial Time ◽

Open Problem ◽

Graph Isomorphism ◽

Polynomial Time Algorithm ◽

Time Algorithm ◽

Isomorphism Problem ◽

Cospectral Graphs ◽

Graph Isomorphism Problem ◽

Spectra Of Graphs ◽

New Construction

The graph isomorphism problem is to check if two given graphs are isomorphic. Graph isomorphism is a well studied problem and numerous algorithms are available for its solution. In this paper we present algorithms for graph isomorphism that employ the spectra of graphs. An open problem that has fascinated many a scientist is if there exists a polynomial time algorithm for graph isomorphism. Though we do not solve this problem in this paper, the algorithms we present take polynomial time. These algorithms have been tested on a good collection of instances. However, we have not been able to prove that our algorithms will work on all possible instances. In this paper, we also give a new construction for cospectral graphs.

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The Decision-Making Neural Networks Model for Solving the Graph Isomorphism Problem

Chinese Journal of Computers ◽

10.3724/sp.j.1016.2010.00300 ◽

2010 ◽

Vol 33 (2) ◽

pp. 300-304 ◽

Cited By ~ 1

Author(s):

Jin-Hua NAN ◽

Huan QI

Keyword(s):

Neural Networks ◽

Decision Making ◽

Graph Isomorphism ◽

Isomorphism Problem ◽

Graph Isomorphism Problem

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MET: a Java package for fast molecule equivalence testing

Journal of Cheminformatics ◽

10.1186/s13321-020-00480-1 ◽

2020 ◽

Vol 12 (1) ◽

Author(s):

Jördis-Ann Schüler ◽

Steffen Rechner ◽

Matthias Müller-Hannemann

Keyword(s):

Chemical Properties ◽

Graph Isomorphism ◽

Isomorphism Problem ◽

Equivalence Classes ◽

Equivalence Testing ◽

Equivalence Test ◽

Graph Isomorphism Problem ◽

2D Structure ◽

Node Labels ◽

Labelled Graphs

AbstractAn important task in cheminformatics is to test whether two molecules are equivalent with respect to their 2D structure. Mathematically, this amounts to solving the graph isomorphism problem for labelled graphs. In this paper, we present an approach which exploits chemical properties and the local neighbourhood of atoms to define highly distinctive node labels. These characteristic labels are the key for clever partitioning molecules into molecule equivalence classes and an effective equivalence test. Based on extensive computational experiments, we show that our algorithm is significantly faster than existing implementations within , and . We provide our Java implementation as an easy-to-use, open-source package (via GitHub) which is compatible with . It fully supports the distinction of different isotopes and molecules with radicals.

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

Communications of the ACM ◽

10.1145/3453943 ◽

2021 ◽

Vol 64 (5) ◽

pp. 98-105

Author(s):

Martin Grohe ◽

Daniel Neuen

Keyword(s):

Fixed Point ◽

Complexity Theory ◽

Polynomial Time ◽

Graph Isomorphism ◽

Isomorphism Problem ◽

Descriptive Complexity ◽

Graph Isomorphism Problem ◽

Structural Graph ◽

Graph Classes ◽

Bounded Rank

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.

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