Coloring a Dominating Set Without Conflicts: q-Subset Square Coloring
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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 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.
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