Graphs without a partition into two proportionally dense subgraphs

TL;DR

This paper investigates the existence of partitions into two proportionally dense subgraphs in graphs, providing a counterexample class that disproves the conjecture that all graphs (except stars) have such partitions.

Contribution

It introduces a class of graphs that do not admit a 2-PDS partition, resolving an open question from prior research.

Findings

Identified a class of graphs without 2-PDS partitions.

Disproved the conjecture that all graphs (except stars) have 2-PDS partitions.

Provided a negative answer to an open problem in graph theory.

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{$2$-PDS partition}. The question whether all graphs (except stars) have $2$-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 $2$-PDS partition.