For a positive integer $s$, an $s$-club in a graph $G$ is a set of vertices inducing a subgraph with diameter at most $s$. As generalizations of cliques, $s$-clubs offer a flexible model for real-world networks. This paper addresses the problems of partitioning and disjoint covering of vertices with $s$-clubs on bipartite graphs. First we consider the $(k,s)$-PC problem where ask whether the vertices of $G$ can be partitioned into at most $k$ disjoint $s$-clubs. We prove that for any fixed $k \geq 2$ and for any fixed odd $s \geq 3$ or even $s\geq 8$, the $(k,s)$-PC problem is NP-complete even for bipartite graphs. Note that our NP-completeness result is stronger than the one in Abbas and Stewart (1999), as we assume that both $s$ and $k$ are constants and not part of the input. Additionally, we study the Maximum Disjoint $(t,s)$-Club Covering problem ($(t,s)$-MAX-DCC), which aims to find a collection of vertex-disjoint $(t,s)$-clubs (i.e. $s$-clubs with at least $t$ vertices) that covers the maximum number of vertices in $G$. We prove that it is NP-hard to achieve an approximation factor of $\frac{95}{94} $ for $(t,3)$-MAX-DCC for any fixed $t\geq 8$ and for $(t,2)$-MAX-DCC for any fixed $t\geq 5$ even for bipartite graphs. Previously, results were known only for $(3,2)$-MAX-DCC. Finally, we provide a polynomial-time algorithm for $(2,2)$-MAX-DCC resolving an open problem from Dondi \textit{et al.} (2019).

翻译：对于正整数$s$，图$G$中的$s$-团是指一个顶点子集，其所诱导的子图直径至多为$s$。作为团（clique）的推广，$s$-团为实际网络提供了灵活的建模方式。本文研究二分图上利用$s$-团对顶点进行划分与不相交覆盖的问题。首先考虑$(k,s)$-划分问题（PC问题），即询问$G$的顶点能否被划分为至多$k$个不相交的$s$-团。我们证明，对于任意固定的$k \geq 2$，以及任意固定的奇数$s \geq 3$或偶数$s \geq 8$，即使是在二分图上，$(k,s)$-PC问题也是NP完全的。注意，我们的NP完全性结果比Abbas和Stewart（1999）的结果更强，因为我们假设$s$和$k$均为常数而非输入的一部分。此外，我们研究了最大不相交$(t,s)$-团覆盖问题（$(t,s)$-MAX-DCC问题），其目标是找到一组顶点不相交的$(t,s)$-团（即包含至少$t$个顶点的$s$-团），使得覆盖的顶点数最大化。我们证明，对于任意固定的$t \geq 8$时的$(t,3)$-MAX-DCC问题以及任意固定的$t \geq 5$时的$(t,2)$-MAX-DCC问题，即使是在二分图上，要达到$\frac{95}{94}$的近似因子也是NP困难的。此前，相关结果仅针对$(3,2)$-MAX-DCC问题成立。最后，我们给出了$(2,2)$-MAX-DCC问题的多项式时间算法，解决了Dondi等人（2019）提出的一个开放问题。