In this paper, we consider the following two problems: (i) Deletion Blocker($\alpha$) where we are given an undirected graph $G=(V,E)$ and two integers $k,d\geq 1$ and ask whether there exists a subset of vertices $S\subseteq V$ with $|S|\leq k$ such that $\alpha(G-S) \leq \alpha(G)-d$, that is the independence number of $G$ decreases by at least $d$ after having removed the vertices from $S$; (ii) Transversal($\alpha$) where we are given an undirected graph $G=(V,E)$ and two integers $k,d\geq 1$ and ask whether there exists a subset of vertices $S\subseteq V$ with $|S|\leq k$ such that for every maximum independent set $I$ we have $|I\cap S| \geq d$. We show that both problems are polynomial-time solvable in the class of co-comparability graphs by reducing them to the well-known Vertex Cut problem. Our results generalize a result of [Chang et al., Maximum clique transversals, Lecture Notes in Computer Science 2204, pp. 32-43, WG 2001] and a recent result of [Hoang et al., Assistance and interdiction problems on interval graphs, Discrete Applied Mathematics 340, pp. 153-170, 2023].

翻译：本文研究以下两个问题：(i) 删除阻断器($\alpha$)问题：给定无向图$G=(V,E)$及两个整数$k,d\geq 1$，询问是否存在子集$S\subseteq V$且$|S|\leq k$，使得$\alpha(G-S) \leq \alpha(G)-d$，即移除$S$中顶点后图的独立数至少减少$d$；(ii) 横贯($\alpha$)问题：给定无向图$G=(V,E)$及两个整数$k,d\geq 1$，询问是否存在子集$S\subseteq V$且$|S|\leq k$，使得对每个最大独立集$I$均有$|I\cap S| \geq d$。我们通过将这两个问题归约到经典顶点割问题，证明它们在共可比图类中均可在多项式时间内求解。该结果推广了[Chang等，最大团横贯，计算机科学讲义2204，第32-43页，WG 2001]的结论以及[Hoang等，区间图上的辅助与拦截问题，离散应用数学340，第153-170页，2023]的最新成果。