## A KURATOWSKI THEOREM FOR ORIENTABLE SURFACES

dc.contributor.author | Vollmerhaus, Walter | eng |

dc.date.accessioned | 2008-02-26T23:03:50Z | |

dc.date.available | 2008-02-26T23:03:50Z | |

dc.date.computerscience | 1999-05-27 | eng |

dc.date.issued | 1987-12-01 | eng |

dc.description.abstract | Let $SIGMA$ denote a 2-dimensional surface. A graph $G$ is irreducible for $SIGMA$ provided that $G$ does not embed into $SIGMA$, but every proper subgraph of $G$ does. Let $I$($SIGMA$) denote the set of graphs with vertex degree at least three that are irreducible for $SIGMA$. In this paper we prove that $I$($SIGMA$) is finite for each orientable surface. Together with the result by D. Archdeacon and Huneke, stating that $I$($SIGMA$) is finite for each non-orientable surface, this settles a conjecture of Erd$o dotdot$'s from the 1930s that $I$($SIGMA$) is finite for each surface $SIGMA$. Let $SIGMA sub n$ denote the closed orientable surface of genus $n$. We also write $gamma$($SIGMA$) to denote the genus of orientable surface $SIGMA$. Let $G$ be a finite graph. An embedding of $G$ into a surface $SIGMA$ is a topological map \o'0 /'$:G -> SIGMA$. The orientable genus $gamma$($G$) of the graph $G$ is defined to be the least value of $gamma$($SIGMA$) for all orientable surfaces $SIGMA$ into which $G$ can be embedded. Let $P$ be a property of a graph $G$. We say that $G$ is $P$-critical provided that $G$ has property $P$ but no proper subgraph of $G$ has property $P$. For example, if $P$ is the property that $gamma$($G$) $>= 1$, then the $P$-critical graphs are the two Kuratowski graphs $K sub 5$ and $K sub 3,3$. In general, if $P$ is the property that $gamma$($G$) $>= n$, then a $P$-critical graph can be embedded in $SIGMA sub n$ but not in $SIGMA sub {n - 1}$. Such a $P$-critical graph is also called irreducible for the surface $SIGMA sub {n - 1}$. For any surface $SIGMA$, let $I$($SIGMA$) denote the set of graphs that have no vertices of degree two and are irreducible for $SIGMA$. | eng |

dc.description.notes | We are currently acquiring citations for the work deposited into this collection. We recognize the distribution rights of this item may have been assigned to another entity, other than the author(s) of the work.If you can provide the citation for this work or you think you own the distribution rights to this work please contact the Institutional Repository Administrator at digitize@ucalgary.ca | eng |

dc.identifier.department | 1987-288-36 | eng |

dc.identifier.doi | http://dx.doi.org/10.11575/PRISM/31105 | |

dc.identifier.uri | http://hdl.handle.net/1880/45689 | |

dc.language.iso | Eng | eng |

dc.publisher.corporate | University of Calgary | eng |

dc.publisher.faculty | Science | eng |

dc.subject | Computer Science | eng |

dc.title | A KURATOWSKI THEOREM FOR ORIENTABLE SURFACES | eng |

dc.type | unknown | |

thesis.degree.discipline | Computer Science | eng |