Proper Interval Vertex Deletion

Abstract

The NP-complete problem Proper Interval Vertex Deletion is to decide whether an input graph on n vertices and m edges can be turned into a proper interval graph by deleting at most k vertices. Van Bevern et al. (In: Proceedings WG 2010. Lecture notes in computer science, vol. 6410, pp. 232---243, 2010) showed that this problem can be solved in $\mathcal {O}((14k +14)^{k+1} kn^{6})$ time. We improve this result by presenting an $\mathcal {O}(6^{k} kn^{6})$ time algorithm for Proper Interval Vertex Deletion. Our fixed-parameter algorithm is based on a new structural result stating that every connected component of a {claw,net,tent,C4,C5,C6}-free graph is a proper circular arc graph, combined with a simple greedy algorithm that solves Proper Interval Vertex Deletion on {claw,net,tent,C4,C5,C6}-free graphs in $\mathcal {O}(n+m)$ time. Our approach also yields a polynomial-time 6-approximation algorithm for the optimization variant of Proper Interval Vertex Deletion.