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.

翻译：图的\textbf{平方染色}是指对图中顶点进行染色，使得任意两个距离不超过2的不同顶点均分配不同颜色。本文首次研究一类相关染色问题——图的\textbf{子集平方染色}。概括而言，图的子集平方染色研究的是用$q$种颜色对图的支配集进行平方染色。其目标是优化使用的颜色数量。该问题同时推广了已被广泛研究的高效支配集问题。我们证明：对所有$q$值，q-子集平方染色问题在二部图和弦图上均为NP困难问题。进一步，我们研究了该问题以若干结构参数为参数时的参数化复杂度。最后，我们给出了某些图类进行子集平方染色所需颜色数量的界。