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kcommunity
Journal article

Finding k-community structures in special graph classes

BP2-STS

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  • 2024
Published in:
  • Discrete Applied Mathematics. - Elsevier BV. - 2024, vol. 359, p. 159-175
Graph partitioning
Algorithms
Complexity
Community structure
Threshold graphs
Forests
English For an integer k ≥ 2, a k-community structure in an undirected graph is a partition of its vertex set into k sets called communities, each of size at least two, such that every vertex of the graph has proportionally at least as many neighbours in its own community as in any other community. In this paper, we give a necessary and sufficient condition for a forest on n vertices to admit a k-community structure. Furthermore, we provide an O(k^2 · n^2)-time algorithm that computes such a k-community structure in a forest, if it exists. These results extend a result of Bazgan et al., 2018. We also show that if communities are allowed to have size one, then every forest with n ≥ k ≥ 2 vertices admits a k-community structure that can be found in time O(k^2 · n^2). We then consider threshold graphs and show that every connected threshold graph admits a 2-community structure if and only if it is not isomorphic to a star; also if such a 2-community structure exists, we explain how to obtain it in linear time. We further describe an infinite family of disconnected threshold graphs, containing exactly one isolated vertex, that do not admit any 2-community structure. Finally, we present a new infinite family of connected graphs that may contain an even or an odd number of vertices without 2-community structures, even if communities are allowed to have size one.
Faculty
Faculté des sciences économiques et sociales et du management
Department
Département d'informatique
Language
  • English
Classification
Computer science and technology
License
CC BY
Open access status
hybrid
Identifiers
Persistent URL
https://folia.unifr.ch/unifr/documents/329097
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