On the $d$-independence number in 1-planar graphs
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arXiv
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| Auteurs principaux: | , , |
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| Format: | Preprint |
| Publié: |
2024
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| _version_ | 1866909377757184000 |
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| author | Biedl, Therese Bose, Prosenjit Miraftab, Babak |
| author_facet | Biedl, Therese Bose, Prosenjit Miraftab, Babak |
| contents | The $d$-independence number of a graph $G$ is the largest possible size of an independent set $I$ in $G$ where each vertex of $I$ has degree at least $d$ in $G$. Upper bounds for the $d$-independence number in planar graphs are well-known for $d=3,4,5$, and can in fact be matched with constructions that actually have minimum degree $d$. In this paper, we explore the same questions for 1-planar graphs, i.e., graphs that can be drawn in the plane with at most one crossing per edge. We give upper bounds for the $d$-independence number for all $d$. Then we give constructions that match the upper bound, and (for small $d$) also have minimum degree $d$. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2411_02686 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | On the $d$-independence number in 1-planar graphs Biedl, Therese Bose, Prosenjit Miraftab, Babak Combinatorics Discrete Mathematics 05C10, 05C62 The $d$-independence number of a graph $G$ is the largest possible size of an independent set $I$ in $G$ where each vertex of $I$ has degree at least $d$ in $G$. Upper bounds for the $d$-independence number in planar graphs are well-known for $d=3,4,5$, and can in fact be matched with constructions that actually have minimum degree $d$. In this paper, we explore the same questions for 1-planar graphs, i.e., graphs that can be drawn in the plane with at most one crossing per edge. We give upper bounds for the $d$-independence number for all $d$. Then we give constructions that match the upper bound, and (for small $d$) also have minimum degree $d$. |
| title | On the $d$-independence number in 1-planar graphs |
| topic | Combinatorics Discrete Mathematics 05C10, 05C62 |
| url | https://arxiv.org/abs/2411.02686 |