Structural and Algorithmic Properties of 2-Community Structures

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.

Locally checkable problems in bounded clique-width graphs

“…The notion of k-community was formally defined in [2]. Given a connected graph G and an integer k ≥ 2, a k-community structure for G is a partition…”

“…The k-community problem consists in deciding whether a given connected graph has a k-community structure. The complexity status of the problem in general graphs is currently unknown, and polynomial algorithms exist only for k = 2 in graphs of maximum degree 3 and, also for k = 2, in graphs of minimum degree |V (G)| − 3 (see [2]). We show that, by using our framework, it can be solved in polynomial time in graphs of bounded clique-width.…”

Graphs without a partition into two proportionally dense subgraphs

“…The proof of the equivalence can be found in [3]. Note that a subgraph containing a single vertex is also a PDS, but obviously a PDS cannot be the entire graph.…”

“…The question about the existence of graphs without a 2-PDS was left open in [3]. To the best of our knowledge, no graphs without a 2-PDS partition were known.…”

“…Now, we present an infinite family of graphs where each graph admits a disconnected 2-PDS partition, but not a connected one. The existence of such graphs was left as an open problem in [3]. G[B], the vertices x and y must be in different parts of the partition.…”