Given n points in the plane and a set of disjoint polygons not containing any vertex in their interior, the set of line segments between the vertices which don't intersect with the polygons can be thought of as a visibility graph. A Polygonal Visibility Representation (PVR) of a graph G is such a representation where the visibility graph obtained is isomorphic to G.
PVR#(G) = the min number of polygons in a PVR of G.
PVR(n) = max PVR(G) over all graphs with n vertices.
Exercise 0: Show that allowing the polygons to have holes does not change the PVR#.
Simple examples: K_n requires 0 polygons, empty graph 1, K_n,n requires 1.
CPVR#(G) is when the polygons are restricted to be convex.
Exercise 1: Find CPVR#(K_n,n).
Thm: There exist graphs which require more than \Omega(n/logn) convex polygons.
Proof: rotate a line around each obstacle and derive a 2n long sequence that encodes the visibility graph generated by it.
Thm: There exist graphs which require more than \Omega(n/logn) polygons (not necessarily convex).
Proof: to be discussed.
Exercise 2: Show that for every h, there exists a graph with PVR# = h.
Exercise 3: Show that one polytope is sufficient for representing any graph in three dimensions.
Homework: Show that in 4 dimensions, 1 convex polytope is sufficient to represent any graph.