steinitz theorems for orthogonal polyhedra
TRANSCRIPT
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Steinitz Theorems for Orthogonal Polyhedra
David Eppsteinand
Elena Mumford
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Steinitz Theorem for Convex Polyhedra
= planar 3-vertex-connected
graphs
Steinitz:
skeletons of convex polyhedra in R3 =
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Simple Orthogonal Polyhedra
• Topology of a sphere
• Simply connected faces
Three mutually perpendicular edges at every vertex
simple orthogonal polyhedra
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Simple Orthogonal Polyhedra
• Topology of a sphere
• Simply connected faces
Three mutually perpendicular edges at every vertex
Orthogonal polyhedra that are NOT simple
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Corner polyhedra
All but 3 faces are oriented towards vector (1,1,1)
= Only three faces are “hidden”
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Corner polyhedra
Hexagonal grid drawings with two bends in total
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XYZ polyhedra
Any axis parallel line contains at most two vertices of the polyhedron
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Skeletons of Simple Orthogonal Polyhedra
are exactlyCubic bipartite planar 2-connected graphs such that the removal of any two vertices leaves at most 2 connected components
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Skeletons of Simple Orthogonal Polyhedra
are exactlyCubic bipartite planar 2-connected graphs such that the removal of any two vertices leaves at most 2 connected components
a graph that is NOTa skeleton of a simple orthogonal polyhedron
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Skeletons of XYZ polyhedra
are exactlycubic bipartite planar 3-connected graphs
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Skeletons of XYZ polyhedra
Eppstein GD‘08A planar graph G is an xyz graph if and only if G is bipartite, cubic, and 3-connected.
are exactlycubic bipartite planar 3-connected graphs
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Skeletons of Corner Polyhedra
are exactlycubic bipartite planar 3-connected graphs s.t. every separating triangle of the planar dual graph has the same parity.
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Skeletons of Corner Polyhedra
are exactlycubic bipartite planar 3-connected graphs s.t. every separating triangle of the planar dual graph has the same parity.
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Skeletons of Corner Polyhedra
are exactlycubic bipartite planar 3-connected graphs s.t. every separating triangle of the planar dual graph has the same parity.
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Skeletons of…
…simple orthogonal polyhedraare cubic bipartite planar 2-connected graphs s.t. the removal of any two vertices leaves at most 2 connected components
…XYZ polyhedracubic bipartite planar 3-connected graphs
…corner polyhedracubic bipartite planar 3-connected graphs s.t. every separating triangle of the planar dual graph has the same parity.
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1. Split the dual along separating triangles
2. Construct polyhedra for 4-connected triangulations3. Glue them together:
Rough outline for a 3-connected graph
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Rooted cycle covers
1. Collection of cycles
2. Every inner vertex is covered exactly once
3. Every white trianglecontains exactly oneedge of the cycle
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Rooted cycle covers
Every 4-connected Eulerian triangulation hasa rooted cycle cover
Rooted cycle cover
embedding as a corner polyhedron
=
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1. Split the dual along separating triangles
2. Construct polyhedra for 4-connected triangulations3. Glue them together:
Rough outline for a 3-connected graph
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Results
• Combinatorial characterizations of skeletons of simple orthogonal polyhedra, corner polyhedraand XYZ polyhedra.
• Algorithms to test a cubic 2-connected graph for being such a skeleton in O(n) randomized expected time or in O(n (log log n)2/log log log n) deterministically with O(n) space.
• Four simple rules to reduce 4-connected Euleriantriangulation to a simpler one while preserving4-connectivity.
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Questions?