  The flip-graph of a convex polygon $\pi$ is the graph whose vertices are the
triangulations of $\pi$ and whose edges correspond to flips between them. The
eccentricity of a triangulation $T$ of $\pi$ is the largest possible distance
in this graph from $T$ to any triangulation of $\pi$. It is well known that,
when all $n-3$ interior edges of $T$ are incident to the same vertex, the
eccentricity of $T$ in the flip-graph of $\pi$ is exactly $n-3$, where $n$
denotes the number of vertices of $\pi$. Here, this statement is generalized to
arbitrary triangulations. Denoting by $n-3-k$ the largest number of interior
edges of $T$ incident to a vertex, it is shown that the eccentricity of $T$ in
the flip-graph of $\pi$ is exactly $n-3+k$, provided $k\leq{n/2-2}$. Inversely,
the eccentricity of a triangulation, when small enough, allows to recover the
value of $k$. More precisely, if $k\leq{n/8-5/2}$, it is also shown that $T$
has eccentricity $n-3+k$ if and only if exactly $n-3-k$ of its interior edges
are incident to a given vertex. When $k>n/2-2$, bounds on the eccentricity of
$T$ are also given and discussed.
