In a vertex-colored graph G=(V,E), a subset S⊆V is said to be consistent if every vertex has a nearest neighbor in S with the same color. The problem of computing a minimum cardinality consistent subset of a graph is known to be NP-hard. On the positive side, Dey et al. (FCT 2021) show that this problem is solvable in polynomial time when input graphs are restricted to bi-colored trees. In this paper, we give a polynomial-time algorithm for this problem on k-colored trees with fixed k.
@article{arxiv.2305.07259,
title = {Minimum Consistent Subset for Trees Revisited},
author = {Hiroki Arimura and Tatsuya Gima and Yasuaki Kobayashi and Hiroomi Nochide and Yota Otachi},
journal= {arXiv preprint arXiv:2305.07259},
year = {2023}
}