[Paper Review] The Steiner 4-diameter of a graph

Zhao Wang, Yaping Mao+2 more|arXiv (Cornell University)|Feb 19, 2017

Graph theory and applications28 references3 citations

TL;DR

This paper characterizes graphs with Steiner 4-diameter equal to 3, 4, or n−1 by analyzing the structure of non-cut vertices and cycle configurations. It establishes that sdiam₄(G) = n−1 if and only if G is isomorphic to one of six specific graph types: T_{a,b,c,d}, △_{a,b,c,d}, △′_{a,b,c,d}, G₁, G₂, or G₃, based on constraints on non-cut vertices and cycle intersections.

ABSTRACT

The Steiner distance of a graph, introduced by Chartrand, Oellermann, Tian and Zou in 1989, is a natural generalization of the concept of classical graph distance. For a connected graph $G$ of order at least $2$ and $S\subseteq V(G)$, the \emph{Steiner distance} $d_G(S)$ among the vertices of $S$ is the minimum size among all connected subgraphs whose vertex sets contain $S$. Let $n,k$ be two integers with $2\leq k\leq n$. Then the \emph{Steiner $k$-eccentricity $e_k(v)$} of a vertex $v$ of $G$ is defined by $e_k(v)=\max \{d(S)\,|\,S\subseteq V(G), \ |S|=k, \ and \ v\in S \}$. Furthermore, the \emph{Steiner $k$-diameter} of $G$ is $sdiam_k(G)=\max \{e_k(v)\,|\, v\in V(G)\}$. In 2011, Chartrand, Okamoto and Zhang showed that $k-1\leq sdiam_k(G)\leq n-1$. In this paper, graphs with $sdiam_4(G)=3,4,n-1$ are characterized, respectively.

Motivation & Objective

• To characterize all connected graphs G of order n with Steiner 4-diameter sdiam₄(G) = 3.
• To characterize all connected graphs G of order n with sdiam₄(G) = 4.
• To characterize all connected graphs G of order n with sdiam₄(G) = n−1, identifying the exact family of graphs achieving this maximum value.
• To establish structural conditions on non-cut vertices and cycle intersections that determine the Steiner 4-diameter.
• To extend prior bounds on Steiner k-diameter by providing exact characterizations for k=4.

Proposed method

• Uses the concept of non-cut vertices and applies an injective mapping from non-cut vertices of a subgraph H to those of the full graph G to preserve structural constraints.
• Applies Corollary 2 and Corollary 3 to bound the number of non-cut vertices based on cycle length and intersection properties.
• Analyzes the structure of G∖V(Cᵢ) for cycles Cᵢ of length 3 or 4, showing it must be a union of pairwise independent paths.
• Employs case analysis based on the circumference c(G), distinguishing between c(G)=3, c(G)=4, and c(G)≥5.
• Uses the fact that sdiam₄(G) ≤ n−2 if c(G) ≥ 5, which eliminates such graphs from achieving maximum diameter.
• Constructs explicit graph families (T_{a,b,c,d}, △_{a,b,c,d}, △′_{a,b,c,d}, G₁, G₂, G₃) to represent all possible graphs with sdiam₄(G) = n−1.

Experimental results

Research questions

• RQ1 Which graphs achieve the minimum possible Steiner 4-diameter, sdiam₄(G) = 3?
• RQ2 Which graphs achieve the next smallest possible Steiner 4-diameter, sdiam₄(G) = 4?
• RQ3 What are the exact structural conditions under which sdiam₄(G) = n−1 for a connected graph G of order n?
• RQ4 How do the number and intersection patterns of cycles influence the Steiner 4-diameter?
• RQ5 What role do non-cut vertices play in determining the Steiner 4-diameter of a graph?

Key findings

• Graphs with sdiam₄(G) = 3 are characterized as those where every 4-subset of vertices has a Steiner tree of size 3, implying a highly centralized structure.
• Graphs with sdiam₄(G) = 4 are characterized by having at most four non-cut vertices and no cycle of length 5 or more.
• The maximum possible Steiner 4-diameter, sdiam₄(G) = n−1, occurs if and only if G is isomorphic to one of six specific graph types: T_{a,b,c,d}, △_{a,b,c,d}, △′_{a,b,c,d}, G₁, G₂, or G₃.
• If G contains a cycle of length at least 5, then sdiam₄(G) ≤ n−2, so such graphs cannot achieve the maximum diameter.
• When G has circumference 4, the number of non-cut vertices is exactly four, and the graph must be isomorphic to G₁, G₂, or G₃.
• For graphs with circumference 3, the structure is determined by the number and intersection pattern of triangles; if there is one triangle, G ≅ △_{a,b,c,d}, and if there are two disjoint or intersecting triangles, G ≅ △′_{a,b,c,d}.