3-trees with few vertices of degree 3 in circuit graphs

Atsuhiro Nakamoto, Yoshiaki Oda, Katsuhiro Ota

Research output: Contribution to journalArticle

10 Citations (Scopus)

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 Fχ with Euler characteristic χ ≥ 0 has a 3-tree with at most frac(n, 3) + cχ vertices of degree 3, where cχ is a constant depending only on Fχ.

Original languageEnglish
Pages (from-to)666-672
Number of pages7
JournalDiscrete Mathematics
Volume309
Issue number4
DOIs
Publication statusPublished - 2009 Mar 6

Fingerprint

Connected graph
Plane Graph
Disjoint Paths
Networks (circuits)
Euler Characteristic
Graph in graph theory
Spanning tree
Maximum Degree
Cycle
Vertex of a graph

Keywords

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

ASJC Scopus subject areas

  • Discrete Mathematics and Combinatorics
  • Theoretical Computer Science

Cite this

3-trees with few vertices of degree 3 in circuit graphs. / Nakamoto, Atsuhiro; Oda, Yoshiaki; Ota, Katsuhiro.

In: Discrete Mathematics, Vol. 309, No. 4, 06.03.2009, p. 666-672.

Research output: Contribution to journalArticle

@article{d92dc366794d4275babed9bbc02cc98e,
title = "3-trees with few vertices of degree 3 in circuit graphs",
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 Fχ with Euler characteristic χ ≥ 0 has a 3-tree with at most frac(n, 3) + cχ vertices of degree 3, where cχ is a constant depending only on Fχ.",
keywords = "3-connected graph, 3-tree, Circuit graph, Surface",
author = "Atsuhiro Nakamoto and Yoshiaki Oda and Katsuhiro Ota",
year = "2009",
month = "3",
day = "6",
doi = "10.1016/j.disc.2008.01.002",
language = "English",
volume = "309",
pages = "666--672",
journal = "Discrete Mathematics",
issn = "0012-365X",
publisher = "Elsevier",
number = "4",

}

TY - JOUR

T1 - 3-trees with few vertices of degree 3 in circuit graphs

AU - Nakamoto, Atsuhiro

AU - Oda, Yoshiaki

AU - Ota, Katsuhiro

PY - 2009/3/6

Y1 - 2009/3/6

N2 - 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 Fχ with Euler characteristic χ ≥ 0 has a 3-tree with at most frac(n, 3) + cχ vertices of degree 3, where cχ is a constant depending only on Fχ.

AB - 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 Fχ with Euler characteristic χ ≥ 0 has a 3-tree with at most frac(n, 3) + cχ vertices of degree 3, where cχ is a constant depending only on Fχ.

KW - 3-connected graph

KW - 3-tree

KW - Circuit graph

KW - Surface

UR - http://www.scopus.com/inward/record.url?scp=60149097255&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=60149097255&partnerID=8YFLogxK

U2 - 10.1016/j.disc.2008.01.002

DO - 10.1016/j.disc.2008.01.002

M3 - Article

VL - 309

SP - 666

EP - 672

JO - Discrete Mathematics

JF - Discrete Mathematics

SN - 0012-365X

IS - 4

ER -