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,..., c_{\alpha}}. We take a subset S of V, such that for every vertex v in V\S, 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^{\prime} of V, such that for every vertex v in V\S^{\prime}, all the vertices in its set of nearest neighbors in S have the same color as v. We refer to such a S^{\prime} 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.

翻译：在连通简单图 G = (V,E) 中，V 中的每个顶点由颜色集 C={c_1, c_2,..., c_{\alpha}} 中的一种颜色着色。我们选取 V 的一个子集 S，使得对于 V\S 中的每个顶点 v，在其 S 中的最近邻集合内至少存在一个与其颜色相同的顶点。我们将这样的 S 称为一致子集（CS）。最小一致子集（MCS）问题即计算最小规模的一致子集。已有研究证明，对于一般图（包括平面图），MCS 是 NP 完全的。我们将研究扩展到区间图和圆图，以期全面理解 MCS 问题在不同图类中的计算复杂性。严格一致子集是一致子集问题的一个变体。我们选取 V 的一个子集 S′，使得对于 V\S′ 中的每个顶点 v，其 S 中最近邻集合内的所有顶点均与 v 颜色相同。我们将这样的 S′ 称为严格一致子集（SCS）。最小严格一致子集（MSCS）问题即计算最小规模的严格一致子集。我们证明了在一般图中 MSCS 是 NP 难的。我们在树中给出了一个 2 近似算法。随后，我们提出了树中的多项式时间算法。接着，我们展示了在路径、蜘蛛图和梳状图中更快的多项式时间算法。