[Paper Review] Split Domination, Independence, and Irredundance in Graphs

Stephen T. Hedetniemi, Fiona Knoll+1 more|arXiv (Cornell University)|May 6, 2016

Advanced Graph Theory Research8 references3 citations

TL;DR

This paper introduces and analyzes split domination, split independence, and split irredundance in graphs by combining domination theory with graph connectivity. It defines new parameters such as split domination number γₛ(G) and split independence number iₛ(G), establishes a split domination chain analogous to the classical domination inequality chain, and proves that connectivity k(G) serves as a lower bound for all split parameters. The key contribution is the formalization and analysis of these split variants, with exact values derived for paths, cycles, and complete bipartite graphs.

ABSTRACT

In 1978, Kulli and Janakiram \citep{KulliJanakiramSplit} defined the split dominating set: a dominating set $S$ of vertices in a graph $G = (V, E)$ is called {\em split dominating} if the induced subgraph $\langle V \setminus S angle$ is either disconnected or a $K_1$. In this paper we introduce the properties split independence and split irredundance. A set $S$ of vertices in a graph $G =(V,E)$ is called a {\em split independent set} if $S$ is independent and the induced subgraph $\langle V \setminus S angle$ is either disconnected or a $K_1$. A set $S$ of vertices in a graph $G = (V,E)$ is called a {\em split irredundant set} if for $u \in S$, $u$ has a private neighbor with respect to $V(S)$ and the induced subgraph $\langle V \setminus S angle$ is either disconnected or a $K_1$.

Motivation & Objective

• To formalize and study the concept of split independence and split irredundance as extensions of classical domination theory.
• To investigate how graph connectivity, particularly vertex connectivity k(G), influences domination-related parameters.
• To establish a new inequality chain for split parameters analogous to the classical domination chain.
• To derive exact values and bounds for split parameters in specific graph families such as paths, cycles, and complete bipartite graphs.
• To explore the relationship between split and nonsplit variants of domination, independence, and irredundance.

Proposed method

• Introduce the concept of a split independent set: an independent set S such that the induced subgraph on V⧵S is disconnected or a single vertex.
• Define a split irredundant set as a set where each vertex has a private neighbor and V⧵S induces a disconnected or trivial graph.
• Propose new graph parameters: split domination number γₛ(G), split independence number iₛ(G), split irredundance number irₛ(G), and their upper counterparts.
• Establish a split domination inequality chain: irₛ(G) ≤ γₛ(G) ≤ iₛ(G) ≤ βₛ(G) ≤ Γₛ(G) ≤ IRₛ(G), analogous to the classical domination chain.
• Use structural graph analysis and case-by-case evaluation on standard graph families (paths, cycles, wheels, complete bipartite graphs) to compute exact values.
• Prove that vertex connectivity k(G) is a lower bound for all split parameters: k(G) ≤ irₛ(G), k(G) ≤ γₛ(G), k(G) ≤ iₛ(G).

Experimental results

Research questions

• RQ1 What are the structural and parameter properties of split independent sets in graphs?
• RQ2 How do split irredundant sets differ from standard irredundant sets, and what are their extremal parameters?
• RQ3 Can a split version of the classical domination inequality chain be established, and what is its structure?
• RQ4 What are the exact values of split domination, independence, and irredundance parameters for standard graph families like paths and cycles?
• RQ5 How do split parameters relate to nonsplit parameters and classical domination parameters?

Key findings

• The split domination number γₛ(G) satisfies γ(G) ≤ γₛ(G) and k(G) ≤ γₛ(G), with γₛ(G) ≤ n·Δ(G)/(Δ(G)+1) for graphs of order n.
• For paths Pₙ, irₛ(G) = γₛ(G) = iₛ(G) = ⌈n/3⌉ and βₛ(G) = Γₛ(G) = IRₛ(G) = ⌈n/2⌉.
• For cycles Cₙ, irₛ(G) = γₛ(G) = iₛ(G) = ⌈n/3⌉ and βₛ(G) = Γₛ(G) = IRₛ(G) = ⌊n/2⌋.
• The vertex connectivity k(G) is a lower bound for all split parameters: k(G) ≤ irₛ(G), k(G) ≤ γₛ(G), and k(G) ≤ iₛ(G).
• For complete bipartite graphs Kₘ,ₙ with 2 ≤ m ≤ n, iₙₛ(Kₘ,ₙ) = m−1 and βₙₛ(Kₘ,ₙ) = n−1.
• For wheels Wₙ, γₙₛ(Wₙ) = 1 and iₙₛ(Wₙ) = 1, while IRₙₛ(Wₙ) = IR(Cₙ), indicating strong structural dependence on the cycle component.