On extremal graphs with exactly one Steiner tree connecting any $k$ vertices

TL;DR

This paper determines the maximum number of edges in graphs with n vertices where exactly one Steiner tree connects any k vertices, extending previous work on local connectivity and Steiner trees.

Contribution

It provides exact values and characterizations for the maximum edges in graphs with exactly one Steiner tree connecting any k vertices for specific k values.

Findings

Exact maximum edge counts for k=3,4,n

Characterizations of extremal graphs for these cases

Extension of connectivity research to Steiner trees

Abstract

The problem of determining the largest number $f(n;\bar{\kappa}\leq \ell)$ of edges for graphs with $n$ vertices and maximal local connectivity at most $\ell$ was considered by Bollob\'{a}s. Li et al. studied the largest number $f(n;\bar{\kappa}_3\leq2)$ of edges for graphs with $n$ vertices and at most two internally disjoint Steiner trees connecting any three vertices. In this paper, we further study the largest number $f(n;\bar{\kappa}_k=1)$ of edges for graphs with $n$ vertices and exactly one Steiner tree connecting any $k$ vertices with $k\geq 3$. It turns out that this is not an easy task to finish, not like the same problem for the classical connectivity parameter. We determine the exact values of $f(n;\bar{\kappa}_k=1)$ for $k=3,4,n$, respectively, and characterize the graphs which attain each of these values.