  The \emph{Square Colouring} of a graph $G$ refers to colouring of vertices of
a graph such that any two distinct vertices which are at distance at most two
receive different colours. In this paper, we initiate the study of a related
colouring problem called the \emph{subset square colouring} of graphs. Broadly,
the subset square colouring of a graph studies the square colouring of a
dominating set of a graph using $q$ colours. Here, the aim is to optimize the
number of colours used. This also generalizes the well-studied Efficient
Dominating Set problem. We show that the q-Subset Square Colouring problem is
NP-hard for all values of $q$ even on bipartite graphs and chordal graphs. We
further study the parameterized complexity of this problem when parameterized
by a number of structural parameters. We further show bounds on the number of
colours needed to subset square colour some graph classes.
