Towards transitive-free digraphs

Abstract

In a digraph D, an arc $e=(x,y)$ in D is considered transitive if there is a path from x to y in $D-e$. A digraph is transitive-free if it does not contain any transitive arc. In the Transitive-free Vertex Deletion (TVD) problem, the goal is to find at most k vertices S such that $D-S$ has no transitive arcs. In our work, we study a more general version of the TVD problem, denoted by ℓ-Relaxed Transitive-free Vertex Deletion (ℓ-RTVD), where we look for at most k vertices S such that $D-S$ has no more than ℓ transitive arcs. We explore ℓ-RTVD on various well-known graph classes of digraphs such as directed acyclic graphs (DAGs), planar DAGs, α-bounded digraphs, tournaments, and their multiple generalizations such as in-tournaments, out-tournaments, local tournaments, acyclic local tournaments, and obtain the following results. Although the problem admits polynomial-time algorithms in tournaments, α-bounded digraphs, and acyclic local tournaments for fixed values of ℓ, it remains $\text{NP}$-Hard even in planar DAGs with maximum degree 6. In the parameterized realm, for ℓ-RTVD on in-tournaments and out-tournaments, we obtain polynomial kernels parameterized by $k+\ell$ for bounded independence number. But the problem remains fixed-parameter intractable on DAGs when parameterized by k.