Width Parameterizations for Knot-free Vertex Deletion on Digraphs

Abstract

A knot in a directed graph G is a strongly connected subgraph Q of G with at least two vertices, such that no vertex in V(Q) is an in-neighbor of a vertex in V(G) \ V(Q). Knots are important graph structures, because they characterize the existence of deadlocks in a classical distributed computation model, the so-called OR-model. Deadlock detection is correlated with the recognition of knot-free graphs as well as deadlock resolution is closely related to the Knot-Free Vertex Deletion (KFVD) problem, which consists of determining whether an input graph G has a subset S ⊆ V(G) of size at most k such that G[V \ S ] contains no knot. Because of natural applications in deadlock resolution, KFVD is closely related to Directed Feedback Vertex Set. In this paper we focus on graph width measure parameterizations for KFVD. First, we show that: (i) KFVD parameterized by the size of the solution k is W[1]-hard even when p, the length of a longest directed path of the input graph, as well as κ, its Kenny-width, are bounded by constants, and we remark that KFVD is para-NP-hard even considering many directed width measures as parameters, but in FPT when parameterized by clique-width; (ii) KFVD can be solved in time 2 O(tw) × n, but assuming ETH it cannot be solved in 2 o(tw) × n O(1) , where tw is the treewidth of the underlying undirected graph. Finally, since the size of a minimum directed feedback vertex set (d f v) is an upper bound for the size of a minimum knot-free vertex deletion set, we investigate parameterization by d f v and we show that (iii) KFVD can be solved in FPT-time parameterized by either d f v + κ or d f v + p; and it admits a Turing kernel by the distance to a DAG having an Hamiltonian path (another parameter larger than d f v). Results of (iii) cannot be improved when replacing d f v by k due to (i). ACM Subject ClassificationDesign and analysis of algorithms → Parameterized complexity and exact algorithms; Mathematics of computing → Graph theory