[Paper Review] Determination of the prime bound of a graph

Abderrahim Boussaïri, Pierre Ille|arXiv (Cornell University)|Jan 7, 2013

Graph Labeling and Dimension Problems3 references3 citations

TL;DR

This paper determines the prime bound of a graph—the minimal number of vertices needed to extend a non-prime graph into a prime graph—by analyzing its largest modular clique and stability numbers. It establishes exact formulas for the prime bound based on whether the maximum of these numbers is a power of two or not, resolving a conjecture on the upper bound of the prime extension size.

ABSTRACT

Given a graph $G$, a subset $M$ of $V(G)$ is a module of $G$ if for each $v\in V(G)\setminus M$, $v$ is adjacent to all the elements of $M$ or to none of them. For instance, $V(G)$, $\emptyset$ and $\{v\}$ ($v\in V(G)$) are modules of $G$ called trivial. Given a graph $G$, $ω_M(G)$ (respectively $α_M(G)$) denotes the largest integer $m$ such that there is a module $M$ of $G$ which is a clique (respectively a stable) set in $G$ with $|M|=m$. A graph $G$ is prime if $|V(G)|\geq 4$ and if all its modules are trivial. The prime bound of $G$ is the smallest integer $p(G)$ such that there is a prime graph $H$ with $V(H)\supseteq V(G)$, $H[V(G)]=G$ and $|V(H)\setminus V(G)|=p(G)$. We establish the following. For every graph $G$ such that $\max(α_M(G),ω_M(G))\geq 2$ and $\log_2(\max(α_M(G),ω_M(G)))$ is not an integer, $p(G)=\lceil\log_2(\max(α_M(G),ω_M(G))) ceil$. Then, we prove that for every graph $G$ such that $\max(α_M(G),ω_M(G))=2^k$ where $k\geq 1$, $p(G)=k$ or $k+1$. Moreover $p(G)=k+1$ if and only if $G$ or its complement admits $2^k$ isolated vertices. Lastly, we show that $p(G)=1$ for every non prime graph $G$ such that $|V(G)|\geq 4$ and $α_M(G)=ω_M(G)=1$.

Motivation & Objective

• To determine the minimal number of vertices required to extend a non-prime graph into a prime graph, known as the prime bound.
• To refine the conjecture by Brignall on the upper bound of the prime extension size using modular clique and stability numbers.
• To characterize the prime bound based on the maximum of the modular stability and clique numbers.
• To resolve the conjecture by showing that the prime bound is tightly bounded by logarithmic functions of the modular parameters.

Proposed method

• Introduces the concepts of modular clique number ωₘ(G) and modular stability number αₘ(G), which measure the largest cliques and stable sets that are also modules.
• Uses induction and recursive construction to build prime extensions, ensuring that no nontrivial modules are introduced in the extended graph.
• Applies a 1-extension technique by adding a single vertex a and carefully controlling its adjacency to existing modules to break nontrivial modules.
• Employs a labeling system λ_G(N) to classify modules and ensure that only prime subgraphs are extended, preserving primality.
• Analyzes the structure of minimal non-prime subgraphs and uses the partitioning of vertex sets into modules to control extension behavior.
• Proves that the prime bound p(G) is exactly ⌈log₂(max(αₘ(G), ωₘ(G)))⌉ when this value is not a power of two, and k or k+1 when it is 2ᵏ.

Experimental results

Research questions

• RQ1 What is the minimal number of vertices needed to extend a non-prime graph into a prime graph, given its largest modular clique or stable set?
• RQ2 How does the prime bound depend on the maximum of the modular stability and clique numbers?
• RQ3 When is the prime bound exactly k and when is it k+1 for graphs where max(αₘ(G), ωₘ(G)) = 2ᵏ?
• RQ4 Does the prime bound remain 1 for all non-prime graphs with no nontrivial modules other than singletons and the full set?
• RQ5 Is the conjecture by Brignall on the prime extension size upper bound valid, and can it be tightened using modular parameters?

Key findings

• For every graph G with max(αₘ(G), ωₘ(G)) ≥ 2 and log₂(max(αₘ(G), ωₘ(G))) not an integer, the prime bound is exactly ⌈log₂(max(αₘ(G), ωₘ(G)))⌉.
• When max(αₘ(G), ωₘ(G)) = 2ᵏ for k ≥ 1, the prime bound is k or k+1, with p(G) = k+1 if and only if G or its complement has 2ᵏ isolated vertices.
• For all non-prime graphs G with |V(G)| ≥ 4 and αₘ(G) = ωₘ(G) = 1, the prime bound is exactly 1.
• The prime bound satisfies the inequality ⌈log₂(max(αₘ(G), ωₘ(G)))⌉ ≤ p(G) ≤ ⌈log₂(max(αₘ(G), ωₘ(G)) + 1)⌉ for all graphs with max(αₘ(G), ωₘ(G)) ≥ 2.
• The construction of a prime extension using a single added vertex ensures that all nontrivial modules are destroyed, and the resulting graph is prime.
• The paper confirms Brignall’s conjecture positively by showing that the upper bound in terms of max(α(G), ω(G)) can be refined using modular parameters.