Graphs without a partition into two proportionally dense subgraphs
Discrete Mathematics
2025-01-15 v4 Computational Complexity
Abstract
A \emph{proportionally dense subgraph} (PDS) is an induced subgraph of a graph with the property that each vertex in the PDS is adjacent to proportionally as many vertices in the subgraph as in the rest of the graph. In this paper, we study a partition of a graph into two proportionally dense subgraphs, namely a \emph{-PDS partition}. The question whether all graphs (except stars) have -PDS partition was left open in [Bazgan et al., Algorithmica 80(6) (2018), 1890--1908]. We give a negative answer on that question and present a class of graphs without a -PDS partition.
Cite
@article{arxiv.1806.10963,
title = {Graphs without a partition into two proportionally dense subgraphs},
author = {Cristina Bazgan and Janka Chlebíková and Clément Dallard},
journal= {arXiv preprint arXiv:1806.10963},
year = {2025}
}