A simple greedy algorithm to find a maximal independent set (MIS) in a graph starts with the empty set and visits every vertex, adding it to the set if and only if none of its neighbours are already in the set. In this paper, we consider the generalisation of this MIS algorithm by letting it start with any set of vertices and we prove the hardness of many decision problems related to this generalisation. Our results are based on two main strategies. Firstly, we view the MIS algorithm as a sequential update of a Boolean network, which we refer to as the MIS network, according to a permutation of the vertex set. The set of fixed points of the MIS network corresponds to the set of MIS of the graph. Our generalisation then consists in starting from any configuration and following a sequential update given by a word of vertices. Secondly, we introduce the concept of a colony of a graph, that is a set of vertices that is dominated by an independent set. Deciding whether a set of vertices is a colony is NP-complete; decision problems related to the MIS algorithm will be reduced from the Colony problem. We first show that deciding whether a configuration can reach all maximal independent sets is coNP-complete. Second, we consider so-called fixing words, that allow to reach a MIS for any initial configuration, and fixing permutations, which we call permises; deciding whether a permutation is fixing is coNP-complete. Third, we show that deciding whether a graph has a permis is coNP-hard. Finally, we generalise the MIS algorithm to digraphs. The algorithm then uses the so-called kernel network, whose fixed points are the kernels of the digraph. Deciding whether the kernel network of a given digraph is fixable is coNP-hard, even for digraphs that have a kernel. Alternatively, we introduce two fixable Boolean networks whose sets of fixed points contain all kernels.

翻译：一种在图中寻找最大独立集（MIS）的简单贪心算法从空集开始，逐一访问每个顶点，当且仅当该顶点的邻居均不在当前集合中时才将其加入。本文考虑该MIS算法的推广形式：允许其从任意顶点集合开始，并证明与此推广相关的多个判定问题的难度。我们的结论基于两种主要策略。首先，将MIS算法视为根据顶点集排列对布尔网络（称为MIS网络）的序贯更新，该MIS网络的不动点集对应图的所有MIS。推广后，我们从任意配置出发，并按照顶点词给定的序贯更新路径进行迭代。其次，引入图的"聚落"概念，即被独立集支配的顶点集合。判定顶点集是否为聚落是NP完全的；与MIS算法相关的判定问题将归约到聚落问题。我们首先证明判定某配置能否到达所有最大独立集是coNP完全的。其次，研究"固定词"（允许从任意初始配置到达MIS的顶点词）与"固定排列"（称为许可排列），并证明判定排列是否为固定排列是coNP完全的。第三，证明判定图是否存在许可排列是coNP难的。最后，将MIS算法推广到有向图，此时算法使用所谓的核网络，其不动点即为有向图的核。即使对于存在核的有向图，判定其核网络是否可固定仍为coNP难问题。作为替代，我们引入两种可固定的布尔网络，其不动点集合包含所有核。