Constructing edge-disjoint Steiner paths in lexicographic product networks

Abstract

The concept of pedant tree-connectivity was introduced by Hager in 1985. For a graph G = (V, E) and a set S ⊆ V (G) of at least two vertices, an S-Steiner tree or a Steiner tree connecting S (or simply, an S-tree) is a such subgraph T = (V ′ , E ′ ) of G that is a tree with S ⊆ V ′ . For an S-Steiner tree, if the degree of each vertex in S is equal to one, then this tree is called a pedant S-Steiner tree. Two pedant S-Steiner trees T and T ′ are said to be internally disjoint if E(T ) ∩ E(T ′ ) = ∅ and V (T )∩V (T ′ ) = S. For S ⊆ V (G) and |S| ≥ 2, the local pedant tree-connectivity τ G (S) is the maximum number of internally disjoint pedant S-Steiner trees in G. For an integer k with 2 ≤ k ≤ n, pedant tree k-connectivity is defined as τ k (G) = min{τ G (S) | S ⊆ V (G), |S| = k}. In this paper, we prove that for any two connected graphs G and H, τ 3 (G H) ≥ min{3⌊ τ3(G) 2 ⌋, 3⌊ τ3(H) 2 ⌋}. Moreover, the bound is sharp.

Generalized Connectivity of Graphs

“…Actually, for the pendant tree-connectivity and path-connectivity, there have been many results published recently. For these results we refer the reader to [50,51,98,99,100].…”

Pendant 3-Tree Connectivity of Augmented Cubes

“…). For an integer k with 2 ≤ k ≤ n, the pendant k-tree-connectivity is defined as Recently, Y. Mao [23,24] worked on pendant tree connectivity. It is clear that,…”