On a Combinatorial Approach to Studying the Steiner Diameter of a Graph and Its Line Graph

Abstract

In 1989, Chartrand, Oellermann, Tian and Zou introduced the Steiner distance for graphs. This is a natural generalization of the classical graph distance concept. Let Γ be a connected graph of order at least 2, and S\V(Γ). Then, the minimum size among all the connected subgraphs whose vertex sets contain S is the Steiner distancedΓ(S) among the vertices of S. The Steiner k-eccentricity ek(v) of a vertex v of Γ is defined by ek(v)=max{dΓ(S)|S\V(Γ),|S|=k,andv∈S}, where n and k are two integers, with 2≤k≤n, and the Steiner k-diameter of Γ is defined by sdiamk(Γ)=max{ek(v)|v∈V(Γ)}. In this paper, we present an algorithm to derive the Steiner distance of a graph; in addition, we obtain a relationship between the Steiner k-diameter of a graph and its line graph. We study various properties of the Steiner diameter through a combinatorial approach. Moreover, we characterize graph Γ when sdiamk(Γ) is given, and we determine sdiamk(Γ) for some special graphs. We also discuss some of the applications of Steiner diameter, including one in education networks.