I am wondering if the concept of a _cut locus_ has been defined and explored in discrete graphs, rather than [their usual home on manifolds][1]?

The [Wikipedia definition][2] (which I believe I (co-?)authored) is:

> The cut locus of $S$ is the closure of the set of all points $p\in X$ that have two or more distinct shortest paths in $X$ from $S$ to $p$.

For my application, $S$ is a single vertex $x$ of a graph $G$,
and path length is measured by the number of edges in a path.
One possible defintion is:

> The _cut locus_ $C(x)$ of a vertex $x$ in a graph $G$ is (a) the set of
> all the vertices $v$ that have two or more distinct paths from $x$,
> unioned with (b) all pairs of vertices $(u,v)$—and the edge between
> them—such
> that $u$ and $v$ have distinct shortest paths from $x$ of the same
> length, and $(u,v)$ is an edge of $G$.

This definition is a bit cumbersome, but I want to capture both
even (a) and odd (b) cycles.
Here is an example, with the even-cycle vertices one color, the odd-cycle edges another:
<br />![Cut Locus in Graph][3]<br />
I see two possible interpretations of the phrase "distinct shortest paths":
<ol>
<li> Two paths are <em>distinct</em> if they are not identical.</li>
<li> Two paths are <em>distinct</em> if they are disjoint, except for the start and end vertices.</li>
</ol>

The figure above uses the first definition, whereas the second definition would remove
the two *-ed vertices from the cut locus (because the paths are not identical; rather 
they share interior vertices and/or edges and so they are not disjoint).

Again, my main question is: Has this this or similar notions been
studied?  I am hoping to find theorems in the literature of the form:

> If $G$ satisfies properties $\{ ... \}$, then $C(x)$ satisfies
properties $\{ ... \}$.

For example, under what conditions on $G$ is the cut locus a forest
(i.e., devoid of cycles)?
I know this is a fishing expedition, but: Thanks for any pointers or ideas!


  [1]: https://en.wikipedia.org/wiki/Cut_locus_%28Riemannian_manifold%29
  [2]: https://en.wikipedia.org/wiki/Cut_locus
  [3]: https://i.sstatic.net/OTmNO.jpg