Approximating Modular Decomposition Is Hard

Habib, Michel; Mouatadid, Lalla; Zou, Mengchuan

doi:10.1007/978-3-030-39219-2_5

Cited by 2 publications

References 27 publications

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(α, β)-Modules in Graphs

Habib¹,

Mouatadid²,

Sopena³

et al. 2021

Preprint

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Modular Decomposition focuses on repeatedly identifying a module M (a collection of vertices that shares exactly the same neighbourhood outside of M ) and collapsing it into a single vertex. This notion of exactitude of neighbourhood is very strict, especially when dealing with real world graphs. We study new ways to relax this exactitude condition. However, generalizing modular decomposition is far from obvious. Most of the previous proposals lose algebraic properties of modules and thus most of the nice algorithmic consequences. We introduce the notion of an (α, β)-module, a relaxation that allows a bounded number of errors in each node and maintains some of the algebraic structure. It leads to a new combinatorial decomposition with interesting properties. Among the main results in this work, we show that minimal (α, β)-modules can be computed in polynomial time, and that every graph admits an (α, β)-modular decomposition tree, thus generalizing Gallai's Theorem (which corresponds to the case for α = β = 0). Unfortunately we give evidence that computing such a decomposition tree can be difficult.

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(α, β)-Modules in Graphs

Habib¹,

Mouatadid²,

Sopena³

et al. 2021

Preprint

View full text Add to dashboard Cite

show abstract

Exploration of regularities in bipartite graphs using GEOGEBRA software

Oliva,

Díaz

2024

LatIA

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A classroom proposal is presented to integrate contents of Graph Theory and Linear Algebra in complete bipartite graphs, linking adjacency and Laplacian matrices, the eigenvalues of graphs will be determined, applicable to connectivity concepts. Students will be given exploration activities working with GeoGebra software, starting from several particular cases, with table works and questionnaires to be completed, in order to determine patterns on the eigenvalues of adjacency and Laplacian matrices of complete bipartite graphs. The work with patterns will lead to the generalization process, to abstract properties from observation and experimentation on examples. This learning experience builds bridges between the concrete and the symbolic, and the student is initiated in research

show abstract

Approximating Modular Decomposition Is Hard

Cited by 2 publications

References 27 publications

(α, β)-Modules in Graphs

(α, β)-Modules in Graphs

Exploration of regularities in bipartite graphs using GEOGEBRA software

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