Logical and Structural Approaches to the Graph Isomorphism Problem

2013 ◽  
pp. 42-42
Author(s):  
Martin Grohe
Keyword(s):  
Graph Isomorphism ◽  
Isomorphism Problem ◽  
Graph Isomorphism Problem

scholarly journals MET: a Java package for fast molecule equivalence testing

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.

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

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.

Physically-motivated dynamical algorithms for the graph isomorphism problem

2005 ◽  
Vol 5 (6) ◽  
pp. 492-506
Author(s):  
S.-Y. Shiau ◽  
R. Joynt ◽  
S.N. Coppersmith
Keyword(s):  
Polynomial Time ◽  
Graph Isomorphism ◽  
Polynomial Time Algorithm ◽  
Time Algorithm ◽  
Isomorphism Problem ◽  
Strongly Regular Graphs ◽  
Regular Graphs ◽  
Original Graph ◽  
Graph Isomorphism Problem ◽  
Strongly Regular

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.

scholarly journals Oriented Coloring on Recursively Defined Digraphs

Algorithms ◽  
2019 ◽  
Vol 12 (4) ◽  
pp. 87 ◽  
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.

The Graph Isomorphism Problem

1993 ◽  
Author(s):  
Johannes Köbler ◽  
Uwe Schöning ◽  
Jacobo Torán
Keyword(s):  
Graph Isomorphism ◽  
Isomorphism Problem ◽  
Graph Isomorphism Problem
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