Graphs without 2-community structures

06/28/2018

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by   Cristina Bazgan, et al.

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In the context of community structure detection, we study the existence of a partition of the vertex set of a graph into two parts such that each part is a community, namely a 2-community structure. We use the definition of a community where each vertex of the graph has a larger proportion of neighbors in its community than in the other community. There are few results about this problem, and it was not known if there exist graphs without 2-community structures, except stars. In this paper, we present a class of graphs without 2-community structures and leave some interesting open questions about the problem.