Links in projective planar 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_ | 1866929740821037056 |
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| author | Foisy, Joel Galván, Luis Ángel Topete Knowles, Evan Nolasco, Uriel Alejandro Shen, Yuanyuan Wickham, Lucy |
| author_facet | Foisy, Joel Galván, Luis Ángel Topete Knowles, Evan Nolasco, Uriel Alejandro Shen, Yuanyuan Wickham, Lucy |
| contents | A graph $G$ is nonseparating projective planar if $G$ has a projective planar embedding without a nonsplit link. Nonseparating projective planar graphs are closed under taking minors and are a superclass of projective outerplanar graphs. We partially characterize the minor-minimal separating projective planar graphs by proving that given a minor-minimal nonouter-projective-planar graph $G$, either $G$ is minor-minimal separating projective planar or $G \dot\cup K_{1}$ is minor-minimal weakly separating projective planar, a necessary condition for $G$ to be separating projective planar.
One way to generalize separating projective planar graphs is to consider type I 3-links consisting of two cycles and a pair of vertices. A graph is intrinsically projective planar type I 3-linked (IPPI3L) if its every projective planar embedding contains a nonsplit type I 3-link. We partially characterize minor-minimal IPPI3L graphs by classifying all minor-minimal IPPI3L graphs with three or more components, and finding many others with fewer components. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2206_05758 |
| institution | arXiv |
| publishDate | 2022 |
| record_format | arxiv |
| spellingShingle | Links in projective planar graphs Foisy, Joel Galván, Luis Ángel Topete Knowles, Evan Nolasco, Uriel Alejandro Shen, Yuanyuan Wickham, Lucy Combinatorics Geometric Topology 57M15, 05C10 A graph $G$ is nonseparating projective planar if $G$ has a projective planar embedding without a nonsplit link. Nonseparating projective planar graphs are closed under taking minors and are a superclass of projective outerplanar graphs. We partially characterize the minor-minimal separating projective planar graphs by proving that given a minor-minimal nonouter-projective-planar graph $G$, either $G$ is minor-minimal separating projective planar or $G \dot\cup K_{1}$ is minor-minimal weakly separating projective planar, a necessary condition for $G$ to be separating projective planar. One way to generalize separating projective planar graphs is to consider type I 3-links consisting of two cycles and a pair of vertices. A graph is intrinsically projective planar type I 3-linked (IPPI3L) if its every projective planar embedding contains a nonsplit type I 3-link. We partially characterize minor-minimal IPPI3L graphs by classifying all minor-minimal IPPI3L graphs with three or more components, and finding many others with fewer components. |
| title | Links in projective planar graphs |
| topic | Combinatorics Geometric Topology 57M15, 05C10 |
| url | https://arxiv.org/abs/2206.05758 |