On the algorithmic construction of classifying spaces and the isomorphism problem for biautomatic groups

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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Diamond Subgraphs in the Reduction Graph of a One-Rule String Rewriting System

Fundamenta Informaticae ◽

10.3233/fi-2021-2002 ◽

2021 ◽

Vol 178 (3) ◽

pp. 173-185

Author(s):

Arthur Adinayev ◽

Itamar Stein

Keyword(s):

Isomorphism Problem ◽

Complete Characterization ◽

Hasse Diagram ◽

Subgraph Isomorphism ◽

Rewriting Systems ◽

Rewriting System ◽

String Rewriting ◽

Reduction Graph

In this paper, we study a certain case of a subgraph isomorphism problem. We consider the Hasse diagram of the lattice Mk (the unique lattice with k + 2 elements and one anti-chain of length k) and find the maximal k for which it is isomorphic to a subgraph of the reduction graph of a given one-rule string rewriting system. We obtain a complete characterization for this problem and show that there is a dichotomy. There are one-rule string rewriting systems for which the maximal such k is 2 and there are cases where there is no maximum. No other intermediate option is possible.

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Classifying spaces for families of subgroups for systolic groups

Groups Geometry and Dynamics ◽

10.4171/ggd/461 ◽

2018 ◽

Vol 12 (3) ◽

pp. 1005-1060 ◽

Cited By ~ 4

Author(s):

Damian Osajda ◽

Tomasz Prytuła

Keyword(s):

Classifying Spaces

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Stable decompositions of classifying spaces of finite abelianp-groups

Mathematical Proceedings of the Cambridge Philosophical Society ◽

10.1017/s0305004100065038 ◽

1988 ◽

Vol 103 (3) ◽

pp. 427-449 ◽

Cited By ~ 34

Author(s):

John C. Harris ◽

Nicholas J. Kuhn

Keyword(s):

Finite Group ◽

Cohomology Theory ◽

Classifying Spaces ◽

Classifying Space ◽

Image Position ◽

A Finite Group

LetBGbe the classifying space of a finite groupG. Consider the problem of finding astabledecompositionintoindecomposablewedge summands. Such a decomposition naturally splitsE*(BG), whereE* is any cohomology theory.

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