A Complete Characterisation of Vertex-multiplications of Trees with Diameter 5

Authors

Willie Han Wah Wong, National Institute of Education, Nanyang Technological University of SingaporeFollow
Eng Guan Tay, National Institute of Education, Nanyang Technological University of SingaporeFollow

Abstract

For a connected graph G, let D(G) be the family of strong orientations of G; and for any D ∈ D(G), we denote by d(D) the diameter of D. The orientation number of G is defined as d(G)=min{d(D) | D ∈ 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)={v₁,v₂,… v_n}. For any sequence of n positive integers (s_i), a G vertex-multiplication, denoted by G(s₁, s₂,…s_n), is the graph with vertex set V*=∪_i=1ⁿV_i and edge set E*, where V_i's are pairwise disjoint sets with |V_i|=s_i, for i=1,2,….n; and for any u,v ∈ V*, uv ∈ E* if and only if u ∈ V_i and v ∈V_j for some i,j ∈{1,2,…n} with I ≠ j such that v_i v_j ∈ E(G). They proved a fundamental classification of G vertex-multiplications, with s_i ≥ 2 for all i=1,2,…n, into three classes C₀, C₁ and C₂, and any vertex-multiplication of a tree with diameter at least 3 does not belong to the class C₂. Furthermore, some necessary and sufficient conditions for C₀ 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 C₀ and C₁.

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Recommended Citation

Wong, Willie Han Wah and Tay, Eng Guan (2021) "A Complete Characterisation of Vertex-multiplications of Trees with Diameter 5," Theory and Applications of Graphs: Vol. 8: Iss. 2, Article 6.
DOI: 10.20429/tag.2021.080206
Available at: https://digitalcommons.georgiasouthern.edu/tag/vol8/iss2/6