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`http://hdl.handle.net/10497/23606`

Title: | |

Authors: | |

Keywords: | Optimal orientation Orientation number Vertex-multiplication Tree |

Issue Date: | 2021 |

Citation: | Wong, W. H. W., & Tay, E. G. (2021). A complete characterisation of vertex-multiplications of trees with diameter 5. Theory and Applications of Graphs, 8(2), Article 6. https://doi.org/10.20429/tag.2021.080206 |

Journal: | Theory and Applications of Graphs |

Abstract: | For a connected graph $G$, let $\mathscr{D}(G)$ be the family of strong orientations of $G$; and for any $D\in\mathscr{D}(G)$, we denote by $d(D)$ the diameter of $D$. The $\textit{orientation number}$ of $G$ is defined as $\bar{d}(G)=\min\{d(D)\mid D\in \mathscr{D}(G)\}$. In 2000, Koh and Tay introduced a new family of graphs, $G$ vertex-multiplications, and extended the results on the orientation number of complete $n$-partite graphs. Suppose $G$ has the vertex set $V(G)=\{v_1,v_2,\ldots, v_n\}$. For any sequence of $n$ positive integers $(s_i)$, a $G$ \textit{vertex-multiplication}, denoted by $G(s_1, s_2,\ldots, s_n)$, is the graph with vertex set $V^*=\bigcup_{i=1}^n{V_i}$ and edge set $E^*$, where $V_i$'s are pairwise disjoint sets with $|V_i|=s_i$, for $i=1,2,\ldots,n$; and for any $u,v\in V^*$, $uv\in E^*$ if and only if $u\in V_i$ and $v\in V_j$ for some $i,j\in \{1,2,\ldots, n\}$ with $i\neq j$ such that $v_i v_j\in E(G)$. They proved a fundamental classification of $G$ vertex-multiplications, with $s_i\ge 2$ for all $i=1,2,\ldots, n$, into three classes $\mathscr{C}_0, \mathscr{C}_1$ and $\mathscr{C}_2$, and any vertex-multiplication of a tree with diameter at least 3 does not belong to the class $\mathscr{C}_2$. Furthermore, some necessary and sufficient conditions for $\mathscr{C}_0$ were established for vertex-multiplications of trees with diameter $5$. In this paper, we give a complete characterisation of vertex-multiplications of trees with diameter $5$ in $\mathscr{C}_0$ and $\mathscr{C}_1$. |

Description: | The open access publication is available at: https://doi.org/10.20429/tag.2021.080206 |

URI: | |

ISSN: | 2470-9859 |

DOI: | |

Funding Agency: | Nanyang Technological University, Singapore |

File Permission: | None |

File Availability: | No file |

Appears in Collections: | Journal Articles |

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