In the Minimum Consistent Subset (MCS) problem, we are presented with a connected simple undirected graph $G=(V,E)$, consisting of a vertex set $V$ of size $n$ and an edge set $E$. Each vertex in $V$ is assigned a color from the set $\{1,2,\ldots, c\}$. The objective is to determine a subset $V' \subseteq V$ with minimum possible cardinality, such that for every vertex $v \in V$, at least one of its nearest neighbors in $V'$ (measured in terms of the hop distance) shares the same color as $v$. The decision problem, indicating whether there exists a subset $V'$ of cardinality at most $l$ for some positive integer $l$, is known to be NP-complete even for planar graphs. In this paper, we establish that the MCS problem for trees, when the number of colors $c$ is considered an input parameter, is NP-complete. We propose a fixed-parameter tractable (FPT) algorithm for MCS on trees running in $O(2^{6c}n^6)$ time, significantly improving the currently best-known algorithm whose running time is $O(2^{4c}n^{2c+3})$. In an effort to comprehensively understand the computational complexity of the MCS problem across different graph classes, we extend our investigation to interval graphs. We show that it remains NP-complete for interval graphs, thus enriching graph classes where MCS remains intractable.

翻译：在最小一致子集（MCS）问题中，我们给定一个连通简单无向图 $G=(V,E)$，其中顶点集 $V$ 规模为 $n$，边集为 $E$。每个顶点 $v \in V$ 被赋予集合 $\{1,2,\ldots, c\}$ 中的一种颜色。目标是确定一个基数尽可能小的子集 $V' \subseteq V$，使得对于每个顶点 $v \in V$，至少有一个在 $V'$ 中（按跳数距离度量）的最近邻与 $v$ 颜色相同。该问题的判定形式——即判断是否存在基数不超过 $l$（$l$ 为正整数）的子集 $V'$——即使对于平面图也被证明是NP完全的。本文证明：当颜色数 $c$ 作为输入参数时，树上的MCS问题是NP完全的。我们为树上的MCS问题提出一个运行时间为 $O(2^{6c}n^6)$ 的固定参数可解（FPT）算法，显著改进了当前最佳算法（运行时间 $O(2^{4c}n^{2c+3})$）。为全面理解MCS问题在不同图类中的计算复杂性，我们将研究扩展至区间图，证明该问题在区间图上仍保持NP完全性，从而丰富了MCS问题保持难解性的图类。