Structure and generation of crossing-critical graphs
Fuente:
arXiv
Guardado en:
| Autores principales: | , , |
|---|---|
| Formato: | Preprint |
| Publicado: |
2018
|
| Materias: | |
| Acceso en línea: | |
| Etiquetas: |
Agregar Etiqueta
Sin Etiquetas, Sea el primero en etiquetar este registro!
|
| _version_ | 1866918487158423552 |
|---|---|
| author | Dvořák, Zdeněk Hliněný, Petr Mohar, Bojan |
| author_facet | Dvořák, Zdeněk Hliněný, Petr Mohar, Bojan |
| contents | We study $c$-crossing-critical graphs, which are the minimal graphs that require at least $c$ edge-crossings when drawn in the plane. For $c=1$ there are only two such graphs without degree-2 vertices, $K_5$ and $K_{3,3}$, but for any fixed $c>1$ there exist infinitely many $c$-crossing-critical graphs. It has been previously shown that $c$-crossing-critical graphs have bounded path-width and contain only a bounded number of internally disjoint paths between any two vertices. We expand on these results, providing a more detailed description of the structure of crossing-critical graphs. On the way towards this description, we prove a new structural characterisation of plane graphs of bounded path-width. Then we show that every $c$-crossing-critical graph can be obtained from a $c$-crossing-critical graph of bounded size by replicating bounded-size parts that already appear in narrow "bands" or "fans" in the graph. This also gives an algorithm to generate all the $c$-crossing-critical graphs of at most given order $n$ in polynomial time per each generated graph. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_1803_01931 |
| institution | arXiv |
| publishDate | 2018 |
| record_format | arxiv |
| spellingShingle | Structure and generation of crossing-critical graphs Dvořák, Zdeněk Hliněný, Petr Mohar, Bojan Combinatorics Computational Geometry 05C10 We study $c$-crossing-critical graphs, which are the minimal graphs that require at least $c$ edge-crossings when drawn in the plane. For $c=1$ there are only two such graphs without degree-2 vertices, $K_5$ and $K_{3,3}$, but for any fixed $c>1$ there exist infinitely many $c$-crossing-critical graphs. It has been previously shown that $c$-crossing-critical graphs have bounded path-width and contain only a bounded number of internally disjoint paths between any two vertices. We expand on these results, providing a more detailed description of the structure of crossing-critical graphs. On the way towards this description, we prove a new structural characterisation of plane graphs of bounded path-width. Then we show that every $c$-crossing-critical graph can be obtained from a $c$-crossing-critical graph of bounded size by replicating bounded-size parts that already appear in narrow "bands" or "fans" in the graph. This also gives an algorithm to generate all the $c$-crossing-critical graphs of at most given order $n$ in polynomial time per each generated graph. |
| title | Structure and generation of crossing-critical graphs |
| topic | Combinatorics Computational Geometry 05C10 |
| url | https://arxiv.org/abs/1803.01931 |