Horiyama et al. (AAAI 2024) considered the problem of generating instances with a unique minimum vertex cover under certain conditions. The Pre-assignment for Uniquification of Minimum Vertex Cover problem (shortly PAU-VC) is the problem, for given a graph $G$, to find a minimum set $S$ of vertices in $G$ such that there is a unique minimum vertex cover of $G$ containing $S$. We show that PAU-VC is fixed-parameter tractable parameterized by clique-width, which improves an exponential algorithm for trees given by Horiyama et al. Among natural graph classes with unbounded clique-width, we show that the problem can be solved in linear time on split graphs and unit interval graphs.

翻译：Horiyama等人（AAAI 2024）研究了在特定条件下生成具有唯一最小顶点覆盖的实例问题。最小顶点覆盖唯一化预分配问题（简称PAU-VC）是指：对于给定图$G$，寻找$G$中一个最小顶点子集$S$，使得存在一个包含$S$的$G$的唯一最小顶点覆盖。我们证明了PAU-VC关于团宽参数是固定参数可解的，这改进了Horiyama等人针对树结构提出的指数级算法。在具有无界团宽的自然图类中，我们证明了该问题在分裂图和单位区间图上可在线性时间内求解。