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

Atsuhiro Nakamoto, Yoshiaki Oda, Katsuhiro Ota

Research output: Contribution to journalArticle

11 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

Keywords

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

ASJC Scopus subject areas

  • Theoretical Computer Science
  • Discrete Mathematics and Combinatorics

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