Consider a drawing (in $\mathbb{R}^2$) of a planar graph. (The drawing is given, contrarily to the common setup in graph theory where we are seeking to build a drawing with specific properties.)

For two vertices $u$ and $v$ of the graph let us call the *edge distance* between $u$ and $v$ the length of the shortest path only following edges connecting $u$ and $v$. In other words, it is the shortest path in the graph weighted by the edge lengths.

I am interested in any general results/references that could apply to this (seemingly, natural?) notion. More specifically, I am interested in results applicable to triangulations and convex drawings.

To narrow the scope of the question, let me showcase it by an example of a specific problem:

Consider a triangulated square with side $1$ in $\mathbb{R}^2$. The triangulation is geometric, i.e. it is drawn in $\mathbb{R}^2$, with straight edges and without self-intersections. It can have extra vertices in the interior of the square, but, for simplicity, let it not have extra vertices on the sides (in other words, the original square sides are edges of the triangulation).

**Is it true that, for any vertex of the triangulation, it is at an edge distance of less than $1$ from at least one of the square vertices?**

Thank you!

embeddingof the graph in the plane. A graph with edge lengths is generally called ametric graph. Another closely related topic isquantum graphwhich are metric graphs plus specified boundary conditions on the vertices for a Laplace type of operator on the graph. $\endgroup$