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Sphere-of-Influence Graphs

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Let
S
be a set of vertices chosen from a grid. Given a vertex
V
in
S
, let
N(V)
be the closest neighbor to
V
in
S
and draw a circle with center
V
and radius
|V-N(V)|
. Draw an edge between two vertices
U
and
V
if their circles intersect. In other words, connect
U
and
V
if
|U-V|≤|V-N(V)|+|U-N(U)|
. This gives the sphere-of-influence graph for the given set of vertices.
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