Constructing Internally Disjoint Pendant Steiner Trees in Cartesian Product Networks

TL;DR

This paper investigates the construction of internally disjoint pendant Steiner trees within Cartesian product networks, providing bounds on their connectivity and demonstrating the sharpness of these bounds.

Contribution

It establishes a lower bound for the pedant tree 3-connectivity of Cartesian product graphs based on the connectivity of the factor graphs, extending previous theoretical results.

Findings

Proves a lower bound for $ au_3(G oxempty H)$ in terms of $ au_3(G)$ and $ au_3(H)$

Shows the bound is sharp for all connected graphs G and H

Extends the theory of pedant tree-connectivity to Cartesian product networks

Abstract

The concept of pedant tree-connectivity was introduced by Hager in 1985. For a graph $G=(V,E)$ and a set $S\subseteq V(G)$ of at least two vertices, \emph{an $S$-Steiner tree} or \emph{a Steiner tree connecting $S$} (or simply, \emph{an $S$-tree}) is a such subgraph $T=(V',E')$ of $G$ that is a tree with $S\subseteq V'$. For an $S$-Steiner tree, if the degree of each vertex in $S$ is equal to one, then this tree is called a \emph{pedant $S$-Steiner tree}. Two pedant $S$-Steiner trees $T$ and $T'$ are said to be \emph{internally disjoint} if $E(T)\cap E(T')=\varnothing$ and $V(T)\cap V(T')=S$. For $S\subseteq V(G)$ and $|S|\geq 2$, the \emph{local pedant tree-connectivity} $\tau_G(S)$ is the maximum number of internally disjoint pedant $S$-Steiner trees in $G$. For an integer $k$ with $2\leq k\leq n$, \emph{pedant tree $k$-connectivity} is defined as $\tau_k(G)=\min\{\tau_G(S)\,|\,S\subseteq V(G),|S|=k\}$. In this paper, we prove that for any two connected graphs $G$ and $H$, $\tau_3(G\Box H)\geq \min\{3\lfloor\frac{\tau_3(G)}{2}\rfloor,3\lfloor\frac{\tau_3(H)}{2}\rfloor\}$. Moreover, the bound is sharp.