Let G be a simple connected graph with vertex set V(G) and edge set E(G. Each vertex of V(G) is colored by a color from the set of colors {c_1, c_2,\dots, c_{\alpha}}. We take a subset S of V(G), such that for every vertex v in V(G)\S, at least one vertex of the same color is present in its set of nearest neighbors in S. We refer to such an S as a consistent subset (CS). The Minimum Consistent Subset (MCS) problem is the computation of a consistent subset of the minimum cardinality. It is established that MCS is NP-complete for general graphs, including planar graphs. The strict consistent subset is a variant of consistent subset problems. We take a subset S^{\prime} of V(G), such that for every vertex v in V(G)\S^{\prime}, all the vertices in its set of nearest neighbors in S^{\prime} have the same color as that of v. We refer to such an S^{\prime} as a strict consistent subset (SCS). The Minimum Strict Consistent Subset (MSCS) problem is the computation of a strict consistent subset of the minimum cardinality. We demonstrate that MSCS is NP-hard for general graphs using a reduction from dominating set problems. We construct a 2-approximation algorithm and a polynomial-time algorithm in trees. Lastly, we conclude the faster polynomial-time algorithms in paths, spiders, and combs.

翻译：设 G 为一个简单连通图，其顶点集为 V(G)，边集为 E(G)。V(G) 中的每个顶点均被集合 {c_1, c_2,\dots, c_{\alpha}} 中的一种颜色着色。我们取 V(G) 的一个子集 S，使得对于 V(G)\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。我们称这样的 S 为一致子集（CS）。最小一致子集（MCS）问题即计算具有最小基数的一致子集。已有研究证明，对于一般图（包括平面图），MCS 问题是 NP 完全的。严格一致子集是一致子集问题的一个变体。我们取 V(G) 的一个子集 S^{\prime}，使得对于 V(G)\S^{\prime} 中的每个顶点 v，其 S^{\prime} 中的最近邻集合内的所有顶点均与 v 颜色相同。我们称这样的 S^{\prime} 为严格一致子集（SCS）。最小严格一致子集（MSCS）问题即计算具有最小基数的严格一致子集。我们通过从支配集问题归约，证明了对于一般图，MSCS 问题是 NP 难的。我们构建了一个 2-近似算法以及一个在树中的多项式时间算法。最后，我们总结了在路径、蜘蛛图和梳状图中更快的多项式时间算法。