On the General Position Number of Mycielskian Graphs
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arXiv
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| Main Authors: | , , , |
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| Format: | Preprint |
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2022
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| _version_ | 1866909155039641600 |
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| author | Thomas, Elias John Chandran, Ullas Tuite, James Di Stefano, Gabriele |
| author_facet | Thomas, Elias John Chandran, Ullas Tuite, James Di Stefano, Gabriele |
| contents | The general position problem for graphs was inspired by the no-three-in-line problem from discrete geometry. A set $S$ of vertices of a graph $G$ is a \emph{general position set} if no shortest path in $G$ contains three or more vertices of $S$. The \emph{general position number} of $G$ is the number of vertices in a largest general position set. In this paper we investigate the general position numbers of the Mycielskian of graphs. We give tight upper and lower bounds on the general position number of the Mycielskian of a graph $G$ and investigate the structure of the graphs meeting these bounds. We determine this number exactly for common classes of graphs, including cubic graphs and a wide range of trees. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2203_08170 |
| institution | arXiv |
| publishDate | 2022 |
| record_format | arxiv |
| spellingShingle | On the General Position Number of Mycielskian Graphs Thomas, Elias John Chandran, Ullas Tuite, James Di Stefano, Gabriele Combinatorics 05C12 05C69 The general position problem for graphs was inspired by the no-three-in-line problem from discrete geometry. A set $S$ of vertices of a graph $G$ is a \emph{general position set} if no shortest path in $G$ contains three or more vertices of $S$. The \emph{general position number} of $G$ is the number of vertices in a largest general position set. In this paper we investigate the general position numbers of the Mycielskian of graphs. We give tight upper and lower bounds on the general position number of the Mycielskian of a graph $G$ and investigate the structure of the graphs meeting these bounds. We determine this number exactly for common classes of graphs, including cubic graphs and a wide range of trees. |
| title | On the General Position Number of Mycielskian Graphs |
| topic | Combinatorics 05C12 05C69 |
| url | https://arxiv.org/abs/2203.08170 |