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

Tight Bounds for Potential Maximal Cliques Parameterized by Vertex Cover