Towards Transitive-free Digraphs
Ankit Abhinav, Satyabrata Jana, Abhishek Sahu
TL;DR
Given a digraph $D$, the problem TVD asks to delete at most $k$ vertices so that $D-S$ is transitive-free, while $\ell$-RTVD generalizes this to allow at most $\ell$ transitive arcs. The authors prove $\ell$-RTVD is NP-hard on planar DAGs with maximum degree $6$, via a reduction from Vertex Cover, and provide polynomial-time algorithms for fixed $\ell$ on several classes (tournaments, $\alpha$-bounded digraphs, acyclic local tournaments). They further establish W[1]-hardness for DAGs parameterized by $k$, but show polynomial kernels for in-tournaments and out-tournaments when parameterized by $k+\ell$ and bounded independence number, and obtain FPT results for the $\ell=0$ case on in-/out-tournaments by reduction to 3-Hitting Set. The work advances understanding of transitive structure in directed graphs and offers kernelization and FPT tools relevant to preprocessing for graph drawing and orientation problems. The results illuminate a separation between tractable and intractable cases across graph classes and motivate future work on structural parameters for DAGs.
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 $\ell$-Relaxed Transitive-free Vertex Deletion ($\ell$-RTVD), where we look for at most $k$ vertices $S$ such that $D-S$ has no more than $\ell$ transitive arcs. We explore $\ell$-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 $\ell$, it remains NP-hard even in planar DAGs with maximum degree 6. In the parameterized realm, for $\ell$-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 $k$.