On the hull and interval numbers of oriented graphs
Fuente:
arXiv
Enregistré dans:
| Auteurs principaux: | , , , |
|---|---|
| Format: | Preprint |
| Publié: |
2022
|
| Sujets: | |
| Accès en ligne: | |
| Tags: |
Ajouter un tag
Pas de tags, Soyez le premier à ajouter un tag!
|
| _version_ | 1866917601911767040 |
|---|---|
| author | Araujo, J. Maia, A. K. Medeiros, P. P. Penso, L. |
| author_facet | Araujo, J. Maia, A. K. Medeiros, P. P. Penso, L. |
| contents | In this work, for a given oriented graph $D$, we study its interval and hull numbers, respectively, in the oriented geodetic, P3 and P3* convexities. This last one, we believe to be formally defined and first studied in this paper, although its undirected version is well-known in the literature.
Concerning bounds, for a strongly oriented graph D, and the oriented geodetic convexity, we prove that $ohng(D)\leq m(D)-n(D)+2$ and that there is at least one such that $ohng(D) = m(D)-n(D)$. We also determine exact values for the hull numbers in these three convexities for tournaments, which imply polynomial-time algorithms to compute them. These results allow us to deduce polynomial-time algorithms to compute $ohnp(D)$ when the underlying graph of $D$ is split or cobipartite.
Moreover, we provide a meta-theorem by proving that if deciding whether $oing(D)\leq k$ or $ohng(D)\leq k$ is NP-hard or W[i]-hard parameterized by $k$, for some $i\in\mathbb{Z_+^*}$, then the same holds even if the underlying graph of $D$ is bipartite. Next, we prove that deciding whether $ohnp(D)\leq k$ or $ohnps(D)\leq k$ is W[2]-hard parameterized by $k$, even if $D$ is acyclic and its underlying graph is bipartite; that deciding whether $ohng(D)\leq k$ is W[2]-hard parameterized by $k$, even if $D$ is acyclic; that deciding whether $oinp(D)\leq k$ or $oinps(D)\leq k$ is NP-complete, even if $D$ has no directed cycles and the underlying graph of $D$ is a chordal bipartite graph; and that deciding whether $oinp(D)\leq k$ or $oinps(D)\leq k$ is W[2]-hard parameterized by $k$, even if the underlying graph of $D$ is split.
Finally, also argue that the interval and hull numbers in the oriented P3 and P3* convexities can be computed in cubic time for graphs of bounded clique-width by using Courcelle's theorem. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2210_01598 |
| institution | arXiv |
| publishDate | 2022 |
| record_format | arxiv |
| spellingShingle | On the hull and interval numbers of oriented graphs Araujo, J. Maia, A. K. Medeiros, P. P. Penso, L. Combinatorics Computational Complexity Discrete Mathematics G.2.2 In this work, for a given oriented graph $D$, we study its interval and hull numbers, respectively, in the oriented geodetic, P3 and P3* convexities. This last one, we believe to be formally defined and first studied in this paper, although its undirected version is well-known in the literature. Concerning bounds, for a strongly oriented graph D, and the oriented geodetic convexity, we prove that $ohng(D)\leq m(D)-n(D)+2$ and that there is at least one such that $ohng(D) = m(D)-n(D)$. We also determine exact values for the hull numbers in these three convexities for tournaments, which imply polynomial-time algorithms to compute them. These results allow us to deduce polynomial-time algorithms to compute $ohnp(D)$ when the underlying graph of $D$ is split or cobipartite. Moreover, we provide a meta-theorem by proving that if deciding whether $oing(D)\leq k$ or $ohng(D)\leq k$ is NP-hard or W[i]-hard parameterized by $k$, for some $i\in\mathbb{Z_+^*}$, then the same holds even if the underlying graph of $D$ is bipartite. Next, we prove that deciding whether $ohnp(D)\leq k$ or $ohnps(D)\leq k$ is W[2]-hard parameterized by $k$, even if $D$ is acyclic and its underlying graph is bipartite; that deciding whether $ohng(D)\leq k$ is W[2]-hard parameterized by $k$, even if $D$ is acyclic; that deciding whether $oinp(D)\leq k$ or $oinps(D)\leq k$ is NP-complete, even if $D$ has no directed cycles and the underlying graph of $D$ is a chordal bipartite graph; and that deciding whether $oinp(D)\leq k$ or $oinps(D)\leq k$ is W[2]-hard parameterized by $k$, even if the underlying graph of $D$ is split. Finally, also argue that the interval and hull numbers in the oriented P3 and P3* convexities can be computed in cubic time for graphs of bounded clique-width by using Courcelle's theorem. |
| title | On the hull and interval numbers of oriented graphs |
| topic | Combinatorics Computational Complexity Discrete Mathematics G.2.2 |
| url | https://arxiv.org/abs/2210.01598 |