On extremal graphs with at most two internally disjoint Steiner trees connecting any three vertices

TL;DR

This paper investigates the maximum number of edges in graphs that guarantee the existence of at least two internally disjoint Steiner trees connecting any three vertices, extending classical connectivity problems.

Contribution

It introduces new bounds for the number of edges ensuring multiple internally disjoint Steiner trees connecting larger vertex sets, generalizing previous pairwise connectivity results.

Findings

Established bounds for graphs with guaranteed two internally disjoint Steiner trees for triples of vertices.

Extended classical connectivity problems to Steiner trees connecting sets of size at least three.

Provided theoretical insights into the relationship between edges and Steiner tree connectivity.

Abstract

The problem of determining the smallest number of edges, $h(n;\bar{\kappa}\geq r)$, which guarantees that any graph with $n$ vertices and $h(n;\bar{\kappa}\geq r)$ edges will contain a pair of vertices joined by $r$ internally disjoint paths was posed by Erd\"{o}s and Gallai. Bollob\'{a}s considered the problem of determining the largest number of edges $f(n;\bar{\kappa}\leq \ell)$ for graphs with $n$ vertices and local connectivity at most $\ell$. One can see that $f(n;\bar{\kappa}\leq \ell)= h(n;\bar{\kappa}\geq \ell+1)-1$. These two problems had received a wide attention of many researchers in the last few decades. In the above problems, only pairs of vertices connected by internally disjoint paths are considered. In this paper, we study the number of internally disjoint Steiner trees connecting sets of vertices with cardinality at least 3.