We study the Equitable Connected Partition (ECP for short) problem, where we are given a graph G=(V,E) together with an integer p, and our goal is to find a partition of V into p parts such that each part induces a connected sub-graph of G and the size of each two parts differs by at most 1. On the one hand, the problem is known to be NP-hard in general and W[1]-hard with respect to the path-width, the feedback-vertex set, and the number of parts p combined. On the other hand, fixed-parameter algorithms are known for parameters the vertex-integrity and the max leaf number. As our main contribution, we resolve a long-standing open question [Enciso et al.; IWPEC '09] regarding the parameterisation by the tree-depth of the underlying graph. In particular, we show that ECP is W[1]-hard with respect to the 4-path vertex cover number, which is an even more restrictive structural parameter than the tree-depth. In addition to that, we show W[1]-hardness of the problem with respect to the feedback-edge set, the distance to disjoint paths, and NP-hardness with respect to the shrub-depth and the clique-width. On a positive note, we propose several novel fixed-parameter algorithms for various parameters that are bounded for dense graphs.

翻译：我们研究公平连通划分问题（简称ECP）：给定图G=(V,E)及整数p，目标是寻找V的一个p划分，使得每个部分在G中诱导出连通子图，且任意两部分的大小相差不超过1。一方面，该问题已知为一般NP难问题，且关于路宽、反馈顶点集和部分数p的组合参数具有W[1]-难性。另一方面，关于顶点完整性和最大叶数参数的固定参数算法已有研究。作为主要贡献，我们解决了关于底层图树深参数化的长期悬而未决的问题[Enciso 等；IWPEC '09]。特别地，我们证明ECP关于4-路顶点覆盖数具有W[1]-难性，该结构参数比树深更具限制性。此外，我们证明了问题关于反馈边集、到不交路的距离的W[1]-难性，以及关于灌木深和团宽的NP难性。在积极方面，我们针对密集图中受界的若干参数提出了多种新型固定参数算法。