On blockers and transversals of maximum independent sets in co-comparability graphs

Abstract

In this paper, we consider the following two problems: (i) Deletion Blocker (α) where we are given an undirected graph G=(V,E) and two integers k,d≥1 and ask whether there exists a subset of vertices S⊆V with |S|≤k such that α(G−S)≤α(G)−d, that is the independence number of G decreases by at least d after having removed the vertices from S; (ii) Transversal (α) where we are given an undirected graph G=(V,E) and two integers k,d≥1 and ask whether there exists a subset of vertices S⊆V with |S|≤k such that for every maximum independent set I we have |I∩S|≥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 generalise a result of Chang et al. (2001) and a recent result of Hoang et al. (2023).