Single Exponential FPT Algorithm for Interval Vertex Deletion and Interval Completion Problem

Let G be an input graph with n vertices and m edges and let k be a fixed parameter. We provide a single exponential FPT algorithm with running time O(cn(n+m)), c = min{18, k} that turns graph G into an interval graph by deleting at most k vertices from G. This solves an open problem posed by Marx [19]. We also provide a single exponential FPT algorithm with running time O(cn(n + m)), c = min{17, k} that turns G into an interval graph by adding at most k edges. The first FPT algorithm with run time O(knm) appeared in STOC 2007 [24]. Our algorithm is the the first single exponential FPT algorithm that improves the running time of the previous algorithm. The algorithms are based on a structural decomposition of G into smaller subgraphs when G is free from small interval graph obstructions. The decomposition allows us to manage the search tree more efficiently.