Minimum Consistent Subset of Simple Graph Classes.

In the minimum consistent subset (MCS) problem, a connected simple undirected graph G=(V,E) is given whose each vertex is colored by one of the colors {c1,c2,…,ck}, and the objective is to compute a minimum size subset C⊆V such that for each vertex v∈V, its set of nearest neighbors in C (with respect to the hop-distance) contains at least one vertex of the same color as v. The decision version of the MCS problem is NP-complete for general graphs. Even for planar graphs, the problem remains NP-complete. We will consider some simple graph classes like path, spider, caterpillar and comb; and present polynomial-time algorithms.