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| Auteurs principaux: | , , , |
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| Format: | Preprint |
| Publié: |
2024
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| Accès en ligne: | https://arxiv.org/abs/2401.05696 |
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| _version_ | 1866914637436420096 |
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| author | Iršič, Vesna Klavžar, Sandi Rus, Gregor Tuite, James |
| author_facet | Iršič, Vesna Klavžar, Sandi Rus, Gregor Tuite, James |
| contents | A subset of vertices of a graph $G$ is a general position set if no triple of vertices from the set lie on a common shortest path in $G$. In this paper we introduce the general position polynomial as $\sum_{i \geq 0} a_i x^i$, where $a_i$ is the number of distinct general position sets of $G$ with cardinality $i$. The polynomial is considered for several well-known classes of graphs and graph operations. It is shown that the polynomial is not unimodal in general, not even on trees. On the other hand, several classes of graphs, including Kneser graphs $K(n,2)$, with unimodal general position polynomials are presented. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2401_05696 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | General position polynomials Iršič, Vesna Klavžar, Sandi Rus, Gregor Tuite, James Combinatorics A subset of vertices of a graph $G$ is a general position set if no triple of vertices from the set lie on a common shortest path in $G$. In this paper we introduce the general position polynomial as $\sum_{i \geq 0} a_i x^i$, where $a_i$ is the number of distinct general position sets of $G$ with cardinality $i$. The polynomial is considered for several well-known classes of graphs and graph operations. It is shown that the polynomial is not unimodal in general, not even on trees. On the other hand, several classes of graphs, including Kneser graphs $K(n,2)$, with unimodal general position polynomials are presented. |
| title | General position polynomials |
| topic | Combinatorics |
| url | https://arxiv.org/abs/2401.05696 |