  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.
