2 research outputs found
Width Parameterizations for Knot-free Vertex Deletion on Digraphs
A knot in a directed graph is a strongly connected subgraph of
with at least two vertices, such that no vertex in is an in-neighbor of
a vertex in . 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 {\sc Knot-Free Vertex Deletion (KFVD)} problem, which
consists of determining whether an input graph has a subset of size at most such that contains no knot. In this
paper we focus on graph width measure parameterizations for {\sc KFVD}. First,
we show that: (i) {\sc KFVD} parameterized by the size of the solution is
W[1]-hard even when , the length of a longest directed path of the input
graph, as well as , its Kenny-width, are bounded by constants, and we
remark that {\sc KFVD} is para-NP-hard even considering many directed width
measures as parameters, but in FPT when parameterized by clique-width; (ii)
{\sc KFVD} can be solved in time , but assuming ETH it
cannot be solved in , where is the treewidth of
the underlying undirected graph. Finally, since the size of a minimum directed
feedback vertex set () is an upper bound for the size of a minimum
knot-free vertex deletion set, we investigate parameterization by and we
show that (iii) {\sc KFVD} can be solved in FPT-time parameterized by either
or ; and it admits a Turing kernel by the distance to a DAG
having an Hamiltonian path.Comment: An extended abstract of this paper was published in IPEC 201