Beyond recognizing well-covered graphs
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arXiv
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| Format: | Preprint |
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2024
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| _version_ | 1866913310475026432 |
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| author | Feghali, Carl Marin, Malory Watrigant, Rémi |
| author_facet | Feghali, Carl Marin, Malory Watrigant, Rémi |
| contents | We prove a number of results related to the computational complexity of recognizing well-covered graphs. Let $k$ and $s$ be positive integers and let $G$ be a graph. Then $G$ is said
- $\mathbf{W_k}$ if for any $k$ pairwise disjoint independent vertex sets $A_1, \dots, A_k$ in $G$, there exist $k$ pairwise disjoint maximum independent sets $S_1, \dots,S_k$ in $G$ such that $A_i \subseteq S_i$ for $i \in [k]$.
- $\mathbf{E_s}$ if every independent set in $G$ of size at most $s$ is contained in a maximum independent set in $G$.
Chvátal and Slater (1993) and Sankaranarayana and Stewart (1992) famously showed that recognizing $\mathbf{W_1}$ graphs or, equivalently, well-covered graphs is coNP-complete. We extend this result by showing that recognizing $\mathbf{W_{k+1}}$ graphs in either $\mathbf{W_k}$ or $\mathbf{E_s}$ graphs is coNP-complete. This answers a question of Levit and Tankus (2023) and strengthens a theorem of Feghali and Marin (2024). We also show that recognizing $\mathbf{E_{s+1}}$ graphs is $Θ_2^p$-complete even in $\mathbf{E_s}$ graphs, where $Θ_2^p = \text{P}^{\text{NP}[\log]}$ is the class of problems solvable in polynomial time using a logarithmic number of calls to a SAT oracle. This strengthens a theorem of Bergé, Busson, Feghali and Watrigant (2023). We also obtain the complete picture of the complexity of recognizing chordal $\mathbf{W_k}$ and $\mathbf{E_s}$ graphs which, in particular, simplifies and generalizes a result of Dettlaff, Henning and Topp (2023). |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2404_07853 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Beyond recognizing well-covered graphs Feghali, Carl Marin, Malory Watrigant, Rémi Combinatorics Discrete Mathematics We prove a number of results related to the computational complexity of recognizing well-covered graphs. Let $k$ and $s$ be positive integers and let $G$ be a graph. Then $G$ is said - $\mathbf{W_k}$ if for any $k$ pairwise disjoint independent vertex sets $A_1, \dots, A_k$ in $G$, there exist $k$ pairwise disjoint maximum independent sets $S_1, \dots,S_k$ in $G$ such that $A_i \subseteq S_i$ for $i \in [k]$. - $\mathbf{E_s}$ if every independent set in $G$ of size at most $s$ is contained in a maximum independent set in $G$. Chvátal and Slater (1993) and Sankaranarayana and Stewart (1992) famously showed that recognizing $\mathbf{W_1}$ graphs or, equivalently, well-covered graphs is coNP-complete. We extend this result by showing that recognizing $\mathbf{W_{k+1}}$ graphs in either $\mathbf{W_k}$ or $\mathbf{E_s}$ graphs is coNP-complete. This answers a question of Levit and Tankus (2023) and strengthens a theorem of Feghali and Marin (2024). We also show that recognizing $\mathbf{E_{s+1}}$ graphs is $Θ_2^p$-complete even in $\mathbf{E_s}$ graphs, where $Θ_2^p = \text{P}^{\text{NP}[\log]}$ is the class of problems solvable in polynomial time using a logarithmic number of calls to a SAT oracle. This strengthens a theorem of Bergé, Busson, Feghali and Watrigant (2023). We also obtain the complete picture of the complexity of recognizing chordal $\mathbf{W_k}$ and $\mathbf{E_s}$ graphs which, in particular, simplifies and generalizes a result of Dettlaff, Henning and Topp (2023). |
| title | Beyond recognizing well-covered graphs |
| topic | Combinatorics Discrete Mathematics |
| url | https://arxiv.org/abs/2404.07853 |