A vertex of a connected graph \(G(V,E,\gamma)\) is a cutvertex if and only if there exist two other vertices \(x,y\in V\) such that every path between \(x\) and \(y\), i.e. a path
\[P^k:=x_0x_1\ldots x_{k-1}x_k\]
with \(x_0=x\), \(x_k=y\) passes through \(v\), i.e. \(x_i=v\) for some \(1\le i\le k-1\).
Proofs: 1