Voronoi diagram in a graph Let $G$ be a graph with (positively) weighted edges.
I want to define the Voronoi diagram for a set of nodes/sites $S$, to
associate with a
node $v \in S$
the subgraph $R(v)$ of $G$ induced by all the nodes
strictly closer to $v$ than to any other node
in $S$, measuring the length of a path by the sum of weights on the arcs.
$R(v)$ is $v$'s Voronoi region.
For example, the green nodes below are in $R(v_1)$, and the yellow nodes
are in $R(v_2)$.
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I would like to understand the structure of the Voronoi diagram.
As a start, what does the diagram of two sites $v_1$ and $v_2$ look like,
i.e., what does the 2-site bisector look like (blue in the above example)?
I think of the bisector $B(v_1,v_2)$ as the complement of $R(v_1) \cup R(v_2)$
in $G$.  Here are two specific questions:


  Q1. Is the bisector of two sites connected in some sense?
  
  Q2. Is $R(v)$ convex in the sense that it contains the shortest path between any two
  nodes in $R(v)$?


Surely this has been studied before.  Can anyone provide references/pointers?
Thanks!

Addendum for Suresh's comment:
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Mehlhorn, K.: A faster approximation algorithm for the Steiner problem in graphs.
Information Processing Letters 27, 125–128 (1988)

Erwig, M.: The graph Voronoi diagram with applications. Networks 36(3), 156–163
(2000)

both references copied from

Matthew T. Dickerson, Michael T. Goodrich, Thomas D. Dickerson, Ying Daisy Zhuo: Round-Trip Voronoi Diagrams and Doubling Density in Geographic Networks. Transactions on Computational Science 14: 211-238 (2011)