Steiner 3-diameter, maximum degree and size of a graph

TL;DR

This paper investigates the minimal size of graphs with given order, maximum degree, and Steiner 3-diameter constraints, generalizing classical diameter problems to Steiner distances.

Contribution

It extends the classical diameter problem to Steiner 3-diameter, analyzing the minimal graph size under these new parameters.

Findings

Derived bounds for minimal graph size with Steiner 3-diameter constraints

Generalized classical diameter results to Steiner 3-diameter context

Provided structural insights into graphs meeting the Steiner diameter conditions

Abstract

The Steiner $k$-diameter $sdiam_k(G)$ of a graph $G$, introduced by Chartrand, Oellermann, Tian and Zou in 1989, is a natural generalization of the concept of classical diameter. When $k=2$, $sdiam_2(G)=diam(G)$ is the classical diameter. The problem of determining the minimum size of a graph of order $n$ whose diameter is at most $d$ and whose maximum is $\ell$ was first introduced by Erd\"{o}s and R\'{e}nyi. In this paper, we generalize the above problem for Steiner $k$-diameter, and study the problem of determining the minimum size of a graph of order $n$ whose Steiner $3$-diameter is at most $d$ and whose maximum is at most $\ell$.