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)，其中每个顶点被k种颜色之一着色，考虑V的子集C，使得对于V\C中的每个顶点v，其在C中的最近邻集合至少包含一个与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对于平面图是NP完全的。我们提出一种多项式时间算法，用于在k为常数时寻找k色蜘蛛图的最小一致子集。同时证明MCS在树上仍然是NP完全的。