### Abstract

A circuit graph(G, C) is a 2-connected plane graph G with an outer cycle C such that from each inner vertex v, there are three disjoint paths to C. In this paper, we shall show that a circuit graph with n vertices has a 3-tree (i.e., a spanning tree with maximum degree at most 3) with at most frac(n - 7, 3) vertices of degree 3. Our estimation for the number of vertices of degree 3 is sharp. Using this result, we prove that a 3-connected graph with n vertices on a surface F_{χ} with Euler characteristic χ ≥ 0 has a 3-tree with at most frac(n, 3) + c_{χ} vertices of degree 3, where c_{χ} is a constant depending only on F_{χ}.

Original language | English |
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Pages (from-to) | 666-672 |

Number of pages | 7 |

Journal | Discrete Mathematics |

Volume | 309 |

Issue number | 4 |

DOIs | |

Publication status | Published - 2009 Mar 6 |

### Keywords

- 3-connected graph
- 3-tree
- Circuit graph
- Surface

### ASJC Scopus subject areas

- Theoretical Computer Science
- Discrete Mathematics and Combinatorics

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## Cite this

Nakamoto, A., Oda, Y., & Ota, K. (2009). 3-trees with few vertices of degree 3 in circuit graphs.

*Discrete Mathematics*,*309*(4), 666-672. https://doi.org/10.1016/j.disc.2008.01.002