Structural and algorithmic properties of 2-community structures

Cristina Bazgan (1) , Janka Chlebikova (2) , Thomas Pontoizeau (1)

1 LAMSADE - Laboratoire d'analyse et modélisation de systèmes pour l'aide à la décision
2 University of Portsmouth

We investigate the structural and algorithmic properties of 2-community structures in graphs introduced recently by Olsen (Math Soc Sci 66(3):331–336, 2013). A 2-community structure is a partition of a vertex set into two parts such that for each vertex the numbers of neighbours in/outside its own part and the sizes of the parts are correlated. We show that some well studied graph classes as graphs of maximum degree 3, minimum degree at least |V|−3, trees and also others, have always a 2-community structure. Furthermore, a 2-community structure can be found in polynomial time in all these classes, even with additional request of connectivity in both parts. We introduce a concept of a weak 2-community and prove that in general graphs it is NP-complete to find a balanced weak 2-community structure with or without request for connectivity in both parts. On the other hand, we present a polynomial-time algorithm to solve the problem (without the condition for connectivity of parts) in graphs of degree at most 3.

Mots clés: Graph theory; Complexity; Graph partitioning; Community structure; Clustering; Social networks; Approximation

Domaines: Informatique [cs]; Mathématique discrète [cs.DM]

Cristina Bazgan, Janka Chlebikova, Thomas Pontoizeau. Structural and algorithmic properties of 2-community structures. Algorithmica, 2018, 80 (6), pp.1890-1908. ⟨10.1007/s00453-017-0283-7⟩. ⟨hal-01482319⟩