Width Parameterizations for Knot-free Vertex Deletion on Digraphs
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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. 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 (dfv) is an upper bound for the size of a minimum knot-free vertex deletion set, we investigate parameterization by dfv and we show that (iii) KFVD can be solved in FPT-time parameterized by either dfv+κ or dfv+p; and it admits a Turing kernel by the distance to a DAG having an Hamiltonian path.
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