[Paper Review] On the Number of Minimal Separators in Graphs

Serge Gaspers, Simon Mackenzie|arXiv (Cornell University)|Mar 4, 2015

Optimization and Search Problems4 citations

TL;DR

This paper provides a simplified proof for the upper bound of $ O(\rho^n \cdot n) $ on the number of minimal separators in an $ n $-vertex graph, where $ \rho = \frac{1+\sqrt{5}}{2} \approx 1.618 $, and establishes a new lower bound of $ \omega(1.4521^n) $, improving upon the prior $ \Omega(3^{n/3}) $ bound. The results tighten the gap between known upper and lower bounds and imply a similar improvement for potential maximal cliques.

ABSTRACT

We consider the largest number of minimal separators a graph on n vertices can have at most. We give a new proof that this number is in $O( ((1+\sqrt{5})/2)^n n )$. We prove that this number is in $ω( 1.4521^n )$, improving on the previous best lower bound of $Ω(3^{n/3}) \subseteq ω( 1.4422^n )$. This gives also an improved lower bound on the number of potential maximal cliques in a graph. We would like to emphasize that our proofs are short, simple, and elementary.

Motivation & Objective

• To simplify the existing upper bound proof for the maximum number of minimal separators in graphs.
• To improve the lower bound on the number of minimal separators beyond $ \Omega(3^{n/3}) $.
• To establish a new lower bound for the number of potential maximal cliques in graphs.
• To close the gap between the best-known upper and lower bounds on minimal separators.

Proposed method

• Introduce a novel measure $ \mu(G,d) = |V| - d $ to track the size of components and separator imbalance in $[a,d]$-separations.
• Use a recursive counting argument based on vertex inclusion or exclusion in separators, leveraging the golden ratio $ \rho $.
• Apply the identity $ \rho^{k-1} + \rho^{k-2} = \rho^k $ to bound the recurrence and prove the upper bound.
• Construct a family of graphs $ G_\ell $ from $ \ell $ copies of a base graph $ G_1 $ with 24 layers, using horizontal and vertical choices to increase separator diversity.
• Count minimal $(a,b)$-separators in $ G_1 $ by ensuring each separator selects one vertex from $ \{v_{1,k},v_{2,k},v_{3,k}\} $ and one from $ \{v_{4,k},v_{5,k},v_{6,k}\} $ for $ k \neq j $, yielding $ 3^{2(|J|-1)} $ per layer.
• Combine separators across $ \ell $ copies to derive a lower bound of $ (|J| \cdot 3^{2(|J|-1)})^{\ell} $, which is $ \omega(1.4521^n) $ for $ n = \ell \cdot 6|J| + 2 $.

Experimental results

Research questions

• RQ1 Can the upper bound on the number of minimal separators be proven more simply than existing methods?
• RQ2 What is the best possible lower bound on the number of minimal separators in an $ n $-vertex graph?
• RQ3 Does the number of potential maximal cliques in a graph also grow faster than $ 3^{n/3} $?
• RQ4 Can the gap between known upper and lower bounds for minimal separators be reduced?

Key findings

• The paper establishes a new upper bound of $ O(\rho^n \cdot n) $ on the number of minimal separators in any $ n $-vertex graph, with a simpler proof than previous approaches.
• It presents a new lower bound of $ \omega(1.4521^n) $ for the number of minimal separators, improving upon the prior $ \Omega(3^{n/3}) \subseteq \omega(1.4422^n) $.
• The construction of $ G_\ell $, formed from $ \ell $ copies of a 24-layer base graph, achieves $ \omega(1.4521^n) $ minimal separators for $ n = \ell \cdot 6 \cdot 24 + 2 $.
• The number of minimal separators in $ G_1 $ is at least $ |J| \cdot 3^{2(|J|-1)} > 2.1271 \times 10^{23} $, with $ |J| = 24 $, demonstrating high separator density.
• As a corollary, the number of potential maximal cliques in a graph is also $ \omega(1.4521^n) $, improving the lower bound for this related concept.
• The results close a significant gap between the best-known upper and lower bounds for minimal separators, showing that $ \mathsf{sep}(n) $ is asymptotically larger than $ 3^{n/3} $.