Eccentricities in the flip-graphs of convex polygons

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 $k\le n\mathrm{/}2-2$. Inversely, the eccentricity of a triangulation, when small enough, allows to recover the value of $k$. More precisely, if $k\le n\mathrm{/}8-5\mathrm{/}2$, 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 $k>n\mathrm{/}2-2$, bounds on the eccentricity of $T$ are also given and discussed.