Colouring a dominating set without conflicts: q-Subset Square Colouring

The 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 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 with q=2 is NP-hard even on planar bipartite graphs and the q-Subset Square Colouring problem is NP-hard 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.