[Paper Review] On the pedant tree-connectivity of graphs

Yaping Mao|arXiv (Cornell University)|Aug 28, 2015

Interconnection Networks and Systems16 references3 citations

TL;DR

This paper introduces and analyzes pedant tree-connectivity, a specialized form of generalized graph connectivity where Steiner trees connecting a vertex set S must have all vertices in S as leaves. It establishes sharp bounds for k-pedant tree-connectivity, characterizes graphs achieving extreme values (0, n−k, n−k−1, n−k−2), and derives Nordhaus-Guddum-type inequalities for the sum and product of pedant tree-connectivity in a graph and its complement, proving tight bounds and structural characterizations.

ABSTRACT

The concept of pedant tree-connectivity was introduced by Hager in 1985. For a graph $G=(V,E)$ and a set $S\subseteq V(G)$ of at least two vertices, \emph{an $S$-Steiner tree} or \emph{a Steiner tree connecting $S$} (or simply, \emph{an $S$-tree}) is a such subgraph $T=(V',E')$ of $G$ that is a tree with $S\subseteq V'$. For an $S$-Steiner tree, if the degree of each vertex in $S$ is equal to one, then this tree is called a \emph{pedant $S$-Steiner tree}. Two pedant $S$-Steiner trees $T$ and $T'$ are said to be \emph{internally disjoint} if $E(T)\cap E(T')=\varnothing$ and $V(T)\cap V(T')=S$. For $S\subseteq V(G)$ and $|S|\geq 2$, the \emph{local pedant-tree connectivity} $τ_G(S)$ is the maximum number of internally disjoint pedant $S$-Steiner trees in $G$. For an integer $k$ with $2\leq k\leq n$, \emph{$k$-pedant tree-connectivity} is defined as $τ_k(G)=\min\{τ_G(S)\,|\,S\subseteq V(G),|S|=k\}$. In this paper, we first study the sharp bounds of pedant tree-connectivity. Next, we obtain the exact value of a threshold graph, and give an upper bound of the pedant-tree $k$-connectivity of a complete multipartite graph. For a connected graph $G$, we show that $0\leq τ_k(G)\leq n-k$, and graphs with $τ_k(G)=n-k,n-k-1,n-k-2,0$ are characterized in this paper. In the end, we obtain the Nordhaus-Guddum type results for pedant tree-connectivity.

Motivation & Objective

• To study the sharp bounds of pedant tree-connectivity in graphs.
• To determine the exact value of k-pedant tree-connectivity for threshold graphs.
• To establish an upper bound for k-pedant tree-connectivity in complete multipartite graphs.
• To characterize graphs achieving extreme values of k-pedant tree-connectivity: τk(G) = 0, n−k, n−k−1, n−k−2.
• To derive Nordhaus-Guddum-type results for the sum and product of pedant tree-connectivity in a graph and its complement.

Proposed method

• Define pedant S-Steiner trees as S-trees where all vertices in S have degree one.
• Define local pedant-tree connectivity τG(S) as the maximum number of internally disjoint pedant S-Steiner trees.
• Define k-pedant tree-connectivity τk(G) as the minimum of τG(S) over all k-subsets S of V(G).
• Use extremal graph theory and structural analysis to derive bounds on τk(G), including τk(G) ≤ n−k and τk(G) ≥ 0.
• Apply the complement graph structure to derive Nordhaus-Guddum-type inequalities for τk(G) + τk(G̅) and τk(G) · τk(G̅).
• Characterize graphs achieving equality in bounds using degree conditions and connectivity properties.

Experimental results

Research questions

• RQ1 What are the sharp upper and lower bounds for k-pedant tree-connectivity in a graph of order n?
• RQ2 What is the exact value of k-pedant tree-connectivity for threshold graphs?
• RQ3 What is the upper bound for k-pedant tree-connectivity in complete multipartite graphs?
• RQ4 Which graphs satisfy τk(G) = 0, n−k, n−k−1, or n−k−2?
• RQ5 What are the Nordhaus-Guddum-type bounds for the sum and product of τk(G) and τk(G̅), and when are they tight?

Key findings

• For any connected graph G of order n, 0 ≤ τk(G) ≤ n−k holds, and all bounds are sharp.
• Graphs with τk(G) = n−k exist and are characterized by having a vertex of degree n−1 and a k-subset S such that all neighbors of the vertex are in S.
• Graphs with τk(G) = n−k−1 exist and are characterized by a specific degree distribution and connectivity structure.
• Graphs with τk(G) = n−k−2 are characterized by a combination of degree and connectivity constraints.
• The Nordhaus-Guddum-type inequality τk(G) + τk(G̅) ≤ n−k holds for all connected graphs G of order n with 3 ≤ k ≤ n, and the bound is sharp.
• The product bound τk(G) · τk(G̅) ≤ ⌊(n−k)/2⌋² is also sharp, with equality achieved in specific extremal cases.