Tight Bounds for Potential Maximal Cliques Parameterized by Vertex Cover

TL;DR

This paper establishes tight exponential bounds on the number of potential maximal cliques in graphs with a given vertex cover size, improving understanding of their combinatorial complexity.

Contribution

It provides the first tight bounds on potential maximal cliques parameterized by vertex cover size, resolving an open problem from prior research.

Findings

Upper bound of 4^k + n potential maximal cliques for graphs with vertex cover of size k

Existence of graphs with Ω(4^k) potential maximal cliques matching the upper bound

Extension of previous results by Fomin et al. with tight bounds

Abstract

We show that a graph with $n$ vertices and vertex cover of size $k$ has at most $4^k + n$ potential maximal cliques. We also show that for each positive integer $k$, there exists a graph with vertex cover of size $k$, $O(k^2)$ vertices, and $\Omega(4^k)$ potential maximal cliques. Our results extend the results of Fomin, Liedloff, Montealegre, and Todinca [Algorithmica, 80(4):1146--1169, 2018], who proved an upper bound of $poly(n) 4^k$, but left the lower bound as an open problem.