Towards Transitive-free Digraphs
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arXiv
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| Format: | Preprint |
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2025
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| _version_ | 1866913734869385216 |
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| author | Abhinav, Ankit Jana, Satyabrata Sahu, Abhishek |
| author_facet | Abhinav, Ankit Jana, Satyabrata Sahu, Abhishek |
| contents | 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 $k$. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2503_10541 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Towards Transitive-free Digraphs Abhinav, Ankit Jana, Satyabrata Sahu, Abhishek Discrete Mathematics Data Structures and Algorithms Combinatorics 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 $k$. |
| title | Towards Transitive-free Digraphs |
| topic | Discrete Mathematics Data Structures and Algorithms Combinatorics |
| url | https://arxiv.org/abs/2503.10541 |