For a graph G = (V,E) where each vertex is coloured by one of k colours, consider a subset C of V such that for each vertex v in V\C, its set of nearest neighbours in C contains at least one vertex of the same colour as v. Such a C is called a consistent subset (CS). Computing a consistent subset of the minimum size is called the Minimum Consistent Subset problem (MCS). MCS is known to be NP-complete for planar graphs. We propose a polynomial-time algorithm for finding a minimum consistent subset of a k-chromatic spider graph when k is a constant. We also show MCS remains NP-complete on trees.

翻译：对于图G=(V,E)，若其每个顶点被k种颜色之一着色，考虑子集C⊆V使得对于V\C中的每个顶点v，其在C中的最近邻集合包含至少一个与v同色的顶点。这样的C称为一致子集(CS)。计算最小规模的一致子集称为最小一致子集问题(MCS)。已知MCS在平面图上为NP完全问题。我们提出了一个多项式时间算法，用于在k为常数时寻找k-色蜘蛛图的最小一致子集。同时，我们证明MCS在树上仍为NP完全问题。