Minimum Strict Consistent Subset in Paths, Spiders, Combs and Trees

In a connected simple graph G = (V,E), each vertex of V is colored by a color from the set of colors C={c_1, c_2,..., cα}. We take a subset S of V, such that for every vertex v in V, at least one vertex of the same color is present in its set of nearest neighbors in S. We refer to such a S as a consistent subset (CS) The Minimum Consistent Subset (MCS) problem is the computation of a consistent subset of the minimum size. It is established that MCS is NP-complete for general graphs, including planar graphs. We expand our study to interval graphs and circle graphs in an attempt to gain a complete understanding of the computational complexity of the MCS problem across various graph classes. The strict consistent subset is a variant of consistent subset problems. We take a subset S^{} of V, such that for every vertex v in V^{}, all the vertices in its set of nearest neighbors in S have the same color as v. We refer to such a S^{} as a strict consistent subset (SCS). The Minimum Strict Consistent Subset (MSCS) problem is the computation of a consistent subset of the minimum size. We demonstrate that MSCS is NP-hard in general graphs. We show a 2-approximation in trees. Later, we show polynomial-time algorithms in trees. Later, we demonstrate faster polynomial-time algorithms in paths, spiders, and combs.