For a graph G = (V,E) where each vertex is coloured by one of k colours, consider a subset C of V such that for each vertex v in V\C, its set of nearest neighbours in C contains at least one vertex of the same colour as v. Such a C is called a consistent subset (CS). Computing a consistent subset of the minimum size is called the Minimum Consistent Subset problem (MCS). MCS is known to be NP-complete for planar graphs. We propose a polynomial-time algorithm for finding a minimum consistent subset of a k-chromatic spider graph when k is a constant. We also show MCS remains NP-complete on trees.
@article{arxiv.2303.02337,
title = {Some results on Minimum Consistent Subsets of Trees},
author = {Bubai Manna and Bodhayan Roy},
journal= {arXiv preprint arXiv:2303.02337},
year = {2023}
}