### Abstract

Consider a graph G with vertex set V in which each of the n vertices is assigned a number from the set {1,...,k} for some positive integer k. This assignment φ is a labeling if all k numbers are used. If φ does not assign adjacent vertices the same label, then φ forms a leveling that partitions V into k levels. If G has a planar drawing in which the y-coordinate of all vertices match their labels and edges are drawn strictly y-monotone, then G is level planar. In this paper, we consider the class of level trees that are level planar regardless of their labeling. We call such trees unlabeled level planar (ULP). Our contributions are three-fold. First, we describe which trees are ULP and provide linear-time level planar drawing algorithms for any labeling. Second, we characterize ULP trees in terms of forbidden subtrees so that any other tree must contain a subtree homeomorphic to one of these. Third, we provide a linear-time recognition algorithm for ULP trees.

Original language | English (US) |
---|---|

Pages (from-to) | 704-721 |

Number of pages | 18 |

Journal | Computational Geometry: Theory and Applications |

Volume | 42 |

Issue number | 6-7 |

DOIs | |

State | Published - Aug 2009 |

### Fingerprint

### Keywords

- Graph drawing
- Level planarity
- Simultaneous embedding
- ULP graphs
- Unlabeled level planarity

### ASJC Scopus subject areas

- Computational Theory and Mathematics
- Computer Science Applications
- Computational Mathematics
- Control and Optimization
- Geometry and Topology

### Cite this

*Computational Geometry: Theory and Applications*,

*42*(6-7), 704-721. https://doi.org/10.1016/j.comgeo.2008.12.006

**Characterization of unlabeled level planar trees.** / Estrella-Balderrama, Alejandro; Fowler, J. Joseph; Kobourov, Stephen G.

Research output: Contribution to journal › Article

*Computational Geometry: Theory and Applications*, vol. 42, no. 6-7, pp. 704-721. https://doi.org/10.1016/j.comgeo.2008.12.006

}

TY - JOUR

T1 - Characterization of unlabeled level planar trees

AU - Estrella-Balderrama, Alejandro

AU - Fowler, J. Joseph

AU - Kobourov, Stephen G

PY - 2009/8

Y1 - 2009/8

N2 - Consider a graph G with vertex set V in which each of the n vertices is assigned a number from the set {1,...,k} for some positive integer k. This assignment φ is a labeling if all k numbers are used. If φ does not assign adjacent vertices the same label, then φ forms a leveling that partitions V into k levels. If G has a planar drawing in which the y-coordinate of all vertices match their labels and edges are drawn strictly y-monotone, then G is level planar. In this paper, we consider the class of level trees that are level planar regardless of their labeling. We call such trees unlabeled level planar (ULP). Our contributions are three-fold. First, we describe which trees are ULP and provide linear-time level planar drawing algorithms for any labeling. Second, we characterize ULP trees in terms of forbidden subtrees so that any other tree must contain a subtree homeomorphic to one of these. Third, we provide a linear-time recognition algorithm for ULP trees.

AB - Consider a graph G with vertex set V in which each of the n vertices is assigned a number from the set {1,...,k} for some positive integer k. This assignment φ is a labeling if all k numbers are used. If φ does not assign adjacent vertices the same label, then φ forms a leveling that partitions V into k levels. If G has a planar drawing in which the y-coordinate of all vertices match their labels and edges are drawn strictly y-monotone, then G is level planar. In this paper, we consider the class of level trees that are level planar regardless of their labeling. We call such trees unlabeled level planar (ULP). Our contributions are three-fold. First, we describe which trees are ULP and provide linear-time level planar drawing algorithms for any labeling. Second, we characterize ULP trees in terms of forbidden subtrees so that any other tree must contain a subtree homeomorphic to one of these. Third, we provide a linear-time recognition algorithm for ULP trees.

KW - Graph drawing

KW - Level planarity

KW - Simultaneous embedding

KW - ULP graphs

KW - Unlabeled level planarity

UR - http://www.scopus.com/inward/record.url?scp=84867970587&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=84867970587&partnerID=8YFLogxK

U2 - 10.1016/j.comgeo.2008.12.006

DO - 10.1016/j.comgeo.2008.12.006

M3 - Article

VL - 42

SP - 704

EP - 721

JO - Computational Geometry: Theory and Applications

JF - Computational Geometry: Theory and Applications

SN - 0925-7721

IS - 6-7

ER -