Horiyama et al. (AAAI 2024) studied the problem of generating graph instances that possess a unique minimum vertex cover under specific conditions. Their approach involved pre-assigning certain vertices to be part of the solution or excluding them from it. Notably, for the \textsc{Vertex Cover} problem, pre-assigning a vertex is equivalent to removing it from the graph. Horiyama et al.~focused on maintaining the size of the minimum vertex cover after these modifications. In this work, we extend their study by relaxing this constraint: our goal is to ensure a unique minimum vertex cover, even if the removal of a vertex may not incur a decrease on the size of said cover. Surprisingly, our relaxation introduces significant theoretical challenges. We observe that the problem is $\Sigma^2_P$-complete, and remains so even for planar graphs of maximum degree 5. Nevertheless, we provide a linear time algorithm for trees, which is then further leveraged to show that MU-VC is in \textsf{FPT} when parameterized by the combination of treewidth and maximum degree. Finally, we show that MU-VC is in \textsf{XP} when parameterized by clique-width while it is fixed-parameter tractable (FPT) if we add the size of the solution as part of the parameter.

翻译：堀山等人（AAAI 2024）研究了在特定条件下生成具有唯一最小顶点覆盖的图实例的问题。他们的方法涉及预先指定某些顶点属于解集或将其排除在解集之外。值得注意的是，对于\textsc{顶点覆盖}问题，预先指定一个顶点等价于将其从图中移除。堀山等人关注于在这些修改后保持最小顶点覆盖的规模。在本工作中，我们通过放宽这一约束来扩展他们的研究：我们的目标是确保存在唯一的最小顶点覆盖，即使移除一个顶点可能不会导致该覆盖的规模减小。令人惊讶的是，我们的放宽引入了显著的理论挑战。我们观察到该问题是$\Sigma^2_P$-完全的，并且即使对于最大度为5的平面图也是如此。尽管如此，我们为树结构提供了一个线性时间算法，并进一步利用该算法证明当以树宽和最大度的组合为参数时，MU-VC属于\textsf{FPT}。最后，我们证明当以团宽为参数时，MU-VC属于\textsf{XP}；而如果将解的大小也作为参数的一部分，则它是固定参数可解的（FPT）。