Structural and Algorithmic Properties of 2-Community Structures

Cristina Bazgan,Janka Chlebikova,Thomas Pontoizeau

doi:10.1007/s00453-017-0283-7

Abstract

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.

Highlights

The research around community structures can be seen as a contribution to the wellestablish research of clustering and graph partitioning
We prove that any graph G = (V, E) of minimum degree |V | − 3 has a connected 2-community structure which can be found in polynomial time
First we prove that every graph of maximum degree 3 has a balanced weak 2-community structure that can be found in polynomial time

Summary

Introduction

The research around community structures can be seen as a contribution to the wellestablish research of clustering and graph partitioning. We study the 2-communities problems with additional constraints such as connectivity or equality of sizes for both parts (a balanced partition). It is known that the problem cannot be approximated within any finite factor in polynomial time in general graphs and it remains APX-hard even on trees of constant maximum degree [12] It demonstrates that some graph partitions problems that are related to e.g. balanced communities are hard to solve even for restricted graph classes and indicates hardness of various problems related to a community structure too. Estivill-Castro et al [11] claimed that the problem to find a k-community structure with restriction to all communities to be connected and equal size is NP-complete in general graphs, but polynomially solvable in trees.

Preliminaries

Connected 2-Community Structures in Some Graph Classes

Balanced 2-Community Structure

Conclusion and Open Problems