Isomorphism of graphs of bounded valence can be tested in polynomial time

Luks, Eugene M.

doi:10.1109/sfcs.1980.24

Cited by 126 publications

(129 citation statements)

References 3 publications

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“…The problem of deciding if there exists a relabeling of indices that makes the adjacency matrices of two given graphs coincide is called the graph isomorphism problem8. In general, the recognition of the isomorphism of simple graphs with bounded valences can be carried out in polynomial time9 and of all graphs in moderately exponential time,

O true(e^{\sqrt{n log n}} true)

where n is the number of nodes in the graph.…”

Section: Basic Conceptsmentioning

confidence: 99%

Scaffold Topologies. 1. Exhaustive Enumeration up to Eight Rings

Pollock

Coutsias

Wester

et al. 2008

J. Chem. Inf. Model.

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Mapping the chemical space of small organic molecules is approached from a theoretical graph theory viewpoint, in an effort to begin the systematic exploration of molecular topologies. We present an algorithm for exhaustive generation of scaffold topologies with up to 8 rings, and an efficient comparison method for graphs within this class. This method uses the return index, a topological invariant derived from the adjacency matrix of the graph, as defined below. Furthermore, we describe an algorithm that verifies the adequacy of the comparison method. Applications of this method for chemical space exploration in the context of drug discovery are discussed. The key result is a unique characterization of scaffold topologies, which may lead to more efficient ways to query large chemical databases.

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O true(e^{\sqrt{n log n}} true)

where n is the number of nodes in the graph.…”

Section: Basic Conceptsmentioning

confidence: 99%

Scaffold Topologies. 1. Exhaustive Enumeration up to Eight Rings

Pollock

Coutsias

Wester

et al. 2008

J. Chem. Inf. Model.

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“…When d = 2, the μ -distance [7] and Nakhleh's m metric [8, 9] are such metrics, but they are no longer metrics for d = 3 (Figure 4 in [8]). Actually, we do not even know whether the isomorphism problem for TSTC networks on a given set S of taxa with globally bounded in-degree hybrid nodes (but without bounding the out-degree of the tree nodes; otherwise, Luks' theorem [19] would apply) is always in P, but we conjecture that this is the case.…”

Section: Discussionmentioning

confidence: 99%

The Comparison of Tree-Sibling Time Consistent Phylogenetic Networks Is Graph Isomorphism-Complete

Cardona

Llabrés

Rosselló

et al. 2014

The Scientific World Journal

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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.

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“…Although it is unknown whether there exists a polynomial-time algorithm for general graphs, a polynomial-time algorithm is known for graphs of bounded degree [12], which means that isomorphism of two chemical graphs can be tested in polynomial time. However, this algorithm is not practical because it is based on group theory and the degree of polynomial in the time complexity is high.…”

Section: Comparison Of Chemical Graphsmentioning

confidence: 99%

Comparison and Enumeration of Chemical Graphs

Akutsu

Nagamochi

2013

Computational and Structural Biotechnology Journal

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Chemical compounds are usually represented as graph structured data in computers. In this review article, we overview several graph classes relevant to chemical compounds and the computational complexities of several fundamental problems for these graph classes. In particular, we consider the following problems: determining whether two chemical graphs are identical, determining whether one input chemical graph is a part of the other input chemical graph, finding a maximum common part of two input graphs, finding a reaction atom mapping, enumerating possible chemical graphs, and enumerating stereoisomers. We also discuss the relationship between the fifth problem and kernel functions for chemical compounds.

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Isomorphism of graphs of bounded valence can be tested in polynomial time

Cited by 126 publications

References 3 publications

Scaffold Topologies. 1. Exhaustive Enumeration up to Eight Rings

Scaffold Topologies. 1. Exhaustive Enumeration up to Eight Rings

The Comparison of Tree-Sibling Time Consistent Phylogenetic Networks Is Graph Isomorphism-Complete

Comparison and Enumeration of Chemical Graphs

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