[Paper Review] New bounds on the signed domination numbers of graphs

S. M. Hosseini Moghaddam, Abdollah Khodkar+1 more|arXiv (Cornell University)|Sep 7, 2014

Advanced Graph Theory Research1 references3 citations

TL;DR

This paper establishes new sharp lower and upper bounds for the signed domination number of graphs, particularly proving that the signed domination number of a tree with $ n $ vertices, $ \ell $ leaves, and $ s $ support vertices is at least $ \frac{n + 4 + 2(\ell - s)}{3} $. The authors use limited packing and tuple domination concepts to derive tight bounds, with the tree bound being both theoretically and constructively sharp.

ABSTRACT

In this paper, we study the signed domination numbers of graphs and present new sharp lower and upper bounds for this parameter. As an example, we present a lower bound on signed domination number of trees in terms of the order, leaves and support vertices.

Motivation & Objective

• To derive new sharp lower and upper bounds for the signed domination number $ \gamma_s(G) $ of general graphs.
• To improve existing bounds by incorporating structural parameters such as minimum degree, packing number, and tree-specific parameters like leaves and support vertices.
• To establish a tighter lower bound for the signed domination number of trees that generalizes and improves upon the known $ \frac{n+4}{3} $ bound.
• To demonstrate the sharpness of the proposed bounds through explicit constructions, particularly for complete graphs and specific tree families.

Proposed method

• Utilizes the concept of $ k $-limited packing to bound the signed domination number from above, leveraging the maximum size of such sets.
• Applies $ k $-tuple domination theory to derive lower bounds by relating the signed domination number to the domination number and minimum degree.
• Employs a rooted tree decomposition technique, partitioning the vertex set into $ V^+ $ and $ V^- $, and analyzing the contribution of vertices in $ V^- $ to their positive descendants.
• Introduces sets $ W_0 $ and $ W_1 $ to classify non-root negative vertices based on whether their positive descendants include support vertices.
• Derives lower bounds via inequalities involving $ \lceil \delta^*/2 \rceil $, the minimum degree of the tree, and the counts of leaves and support vertices.
• Uses algebraic manipulation of vertex count identities: $ |V^+| = \frac{n + \gamma_s(G)}{2} $, $ |V^-| = \frac{n - \gamma_s(G)}{2} $, to transform structural bounds into numerical inequalities.

Experimental results

Research questions

• RQ1 What are new sharp upper bounds for the signed domination number in terms of graph order, minimum degree, and packing number?
• RQ2 How can the signed domination number of a tree be bounded more tightly using its leaf and support vertex counts?
• RQ3 Is the known lower bound $ \gamma_s(T) \geq \frac{n+4}{3} $ for trees improvable by incorporating structural parameters like leaves and support vertices?
• RQ4 Under what conditions is the new bound $ \gamma_s(T) \geq \frac{n + 4 + 2(\ell - s)}{3} $ sharp, and how is this verified?

Key findings

• An upper bound for the signed domination number is established: $ \gamma_s(G) \leq n - 2\left\lfloor \frac{2\rho(G) + \delta(G) - 2}{2} \right\rfloor $, which is sharp for complete graphs.
• For any tree $ T $ of order $ n $ with $ \ell $ leaves and $ s $ support vertices, $ \gamma_s(T) \geq \frac{n + 4 + 2(\ell - s)}{3} $, and this bound is sharp.
• The new tree bound improves upon the classical $ \frac{n+4}{3} $ bound by incorporating the difference $ \ell - s $, reflecting structural sparsity.
• The bound $ \gamma_s(T) \geq \frac{(2\lceil \delta^*/2 \rceil - 1)n + 2(\ell - s + 2)}{2\lceil \delta^*/2 \rceil + 1} $ is derived and shown to be tighter than the $ \frac{n+4}{3} $ bound.
• The proof technique using rooted trees and partitioning of $ V^+ $ into descendant sets of negative vertices ensures structural precision in the bound derivation.
• Sharpness of the tree bound is confirmed via construction, particularly for paths and stars, where equality is achieved.