Graphs without a partition into two proportionally dense subgraphs

TL;DR

This paper investigates the existence of a $2$-PDS partition, where a graph is partitioned into two induced proportionally dense subgraphs. It formalizes the $PDS$ concept with the standard inequality $(|V|-1) d_S(u) \\\ge (|S|-1) d(u)$ for all $u \\\in S$ and studies the partition into two such subgraphs. The authors construct infinite families of graphs with no $2$-PDS partition and another family with a disconnected $2$-PDS partition but no connected one, providing the first negative results. The work highlights the impact of connectivity on $PDS$ partitions and motivates future structural and complexity analyses.

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.

Figures (4)

• Figure 1: A schematic representation of a graph in $\mathcal{G}$.
• Figure 2: A planar graph from $\mathcal{G}$ with 10 vertices without a $2$-PDS partition. On the left, its schematic representation as in \ref{['fig:antisocial graph']}; on the right, its planar representation.
• Figure 3: Four graphs with $11$ vertices which do not have a $2$-PDS partition
• Figure 4: A schematic representation of a graph in $\mathcal{H}$.

Theorems & Definitions (8)

• Definition 1
• Definition 2
• Definition 3
• Theorem 1
• proof
• Definition 4
• Theorem 2
• proof