Completely independent Steiner trees and corresponding tree connectivity

TL;DR

This work defines completely independent $S$-Steiner trees (CISSTs) and the generalized $k^*$-connectivity, extending classical concepts of internally disjoint Steiner trees to enhance fault-tolerant, multi-path routing analysis. It provides a sharp, constructive characterization of CISSTs and delivers exact results for complete graphs, showing $\kappa^*_s(K_n)=n-\lceil s/2\rceil$ and related CIST counts, as well as a tight lower bound for complete bipartite graphs $\kappa^*_s(K_{m,n})$. The results are achieved through explicit constructions and yield polynomial-time realizability for these graph classes, offering practical guidance for robust network design. Overall, the paper advances the theoretical toolkit for analyzing resilience in networks via CISSTs and generalized tree connectivity.

Abstract

The $S$-Steiner tree packing problem provides mathematical foundations for optimizing multi-path information transmission, particularly in designing fault-tolerant parallelized routing architectures for massive-scale network infrastructures. In this article, we propose the definitions of completely independent $S$-Steiner trees (CISSTs for short) and generalized $k^*$-connectivity, which generalize the definitions of internally disjoint $S$-Steiner trees and generalized $k$-connectivity. Given a connected graph $G = (V,E)$ and a vertex subset $S\subseteq V, |S|\geq 2,$ an $S$-Steiner tree of $G$ is a subtree in $G$ that spans all nodes in $S.$ The $S$-Steiner trees $T_1,T_2,\cdots, T_k$ of $G$ are completely independent pairwise if for any $1\leq p<q\leq k,$ $E(T_p)\cap E(T_q)=\emptyset$ , $V(T_p)\cap V(T_q)=S,$ and for any two vertices $x_{1},x_{2}$ in $S$, the paths connecting $x_{1}$ and $x_{2}$ in $T_p,T_q$ are pairwise internally disjoint. The packing number of CISSTs, denoted by $κ^*_G(S),$ is the maximum number of CISSTs in $G.$ The generalized $k^*$-connectivity $κ_k^*(G)$ is the minimum $κ_G^*(S)$ for $S$ ranges over all $k$-subsets of $V(G).$ We provide a detailed characterization of CISSTs. Also, we investigate the CISSTs of complete graphs and complete bipartite graphs. Furthermore, we determine the generalized $k^*$-connectivity for complete graphs and give a tight lower bound of the generalized $k^*$-connectivity for complete bipartite graphs.

Figures (6)

• Figure 1: the $3$ CISSTs in graph $G$
• Figure 2: Traditional design with two internally disjoint Steiner trees
• Figure 3: Enhanced design with two CISSTs
• Figure 4: the $S_i$-Steiner tree $T_j$
• Figure 5: the $S_i$-Steiner tree $T_k'$