  A graph is prime if it does not admit a partition $(A,B)$ of its vertex set
such that $\min\{|A|,|B|\} \geq 2$ and the rank of the $A\times B$ submatrix of
its adjacency matrix is at most $1$. A vertex $v$ of a graph is non-essential
if at least two of the three kinds of vertex-minor reductions at $v$ result in
prime graphs.
  In 1994, Allys proved that every prime graph with at least four vertices has
a non-essential vertex unless it is locally equivalent to a cycle graph. We
prove that every prime graph with at least four vertices has at least two
non-essential vertices unless it is locally equivalent to a cycle graph. As a
corollary, we show that for a prime graph $G$ with at least six vertices and a
vertex $x$, there is a vertex $v \ne x$ such that $G \setminus v$ or $G * v
\setminus v$ is prime, unless $x$ is adjacent to all other vertices and $G$ is
isomorphic to a particular graph on odd number of vertices.
  Furthermore, we show that a prime graph with at least four vertices has at
least three non-essential vertices, unless it is locally equivalent to a graph
consisting of at least two internally-disjoint paths between two fixed distinct
vertices having no common neighbors. We also prove analogous results for
pivot-minors.
