On extremal graphs with at most $\ell$ internally disjoint Steiner trees connecting any n-1 vertices

TL;DR

This paper investigates the maximum number of edges in graphs with a bounded generalized local connectivity, providing exact values for certain cases and bounds for general cases, advancing understanding of graph connectivity constraints.

Contribution

It determines exact edge bounds for graphs with maximum generalized local connectivity at most  for specific cases and constructs graphs for bounds in general cases.

Findings

Exact value of f(n;  n,n-1) determined.

Constructed graphs for lower bounds in general cases.

Extended classical connectivity concepts to generalized local connectivity.

Abstract

The concept of maximum local connectivity $\bar {\kappa}$ of a graph was introduced by Bollob\'{a}s. One of the problems about it is to determine the largest number of edges $f(n;\bar{\kappa}\leq \ell)$ for graphs of order $n$ that have local connectivity at most $\ell$. We consider a generalization of the above concept and problem. For $S\subseteq V(G)$ and $|S|\geq 2$, the \emph{generalized local connectivity} $\kappa(S)$ is the maximum number of internally disjoint trees connecting $S$ in $G$. The parameter $\bar{\kappa}_k(G)=max\{\kappa(S)|S\subseteq V(G),|S|=k\}$ is called the \emph{maximum generalized local connectivity} of $G$. This paper it to consider the problem of determining the largest number $f(n;\bar{\kappa}_k\leq \ell)$ of edges for graphs of order $n$ that have maximum generalized local connectivity at most $\ell$. The exact value of $f(n;\bar{\kappa}_k\leq \ell)$ for $k=n,n-1$ is determined. For a general $k$, we construct a graph to obtain a sharp lower bound.