A knot $K$ in a directed graph $D$ is a strongly connected component of size at least two such that there is no arc $(u,v)$ with $u \in V(K)$ and $v\notin V(K)$. Given a directed graph $D=(V,E)$, we study Knot-Free Vertex Deletion (KFVD), where the goal is to remove the minimum number of vertices such that the resulting graph contains no knots. This problem naturally emerges from its application in deadlock resolution since knots are deadlocks in the OR-model of distributed computation. The fastest known exact algorithm in literature for KFVD runs in time $\mathcal{O}^\star(1.576^n)$. In this paper, we present an improved exact algorithm running in time $\mathcal{O}^\star(1.4549^n)$, where $n$ is the number of vertices in $D$. We also prove that the number of inclusion wise minimal knot-free vertex deletion sets is $\mathcal{O}^\star(1.4549^n)$ and construct a family of graphs with $\Omega(1.4422^n)$ minimal knot-free vertex deletion sets

翻译：有向图$D$中的结$K$是一个大小至少为2的强连通分量，且不存在满足$u \in V(K)$和$v\notin V(K)$的弧$(u,v)$。给定有向图$D=(V,E)$，我们研究无结顶点删除问题（KFVD），其目标是最小化删除的顶点数，使得结果图中不含任何结。该问题自然源于分布式计算OR模型中的死锁解析应用，因为结代表死锁。现有文献中KFVD最快的已知精确算法运行时间为$\mathcal{O}^\star(1.576^n)$。本文提出一种改进的精确算法，运行时间为$\mathcal{O}^\star(1.4549^n)$，其中$n$是$D$中的顶点数。我们还证明了包含关系意义下的最小无结顶点删除集的数量为$\mathcal{O}^\star(1.4549^n)$，并构造了一个具有$\Omega(1.4422^n)$个最小无结顶点删除集的图族。