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Polyhedral gluings of outerplanar graphs
David Richter
Western Michigan University
June 12, 2013
Polyhedral gluings of outerplanar graphs David Richter Western - - PowerPoint PPT Presentation
Polyhedral gluings of outerplanar graphs David Richter Western - - PowerPoint PPT Presentation
Polyhedral gluings of outerplanar graphs David Richter Western Michigan University June 12, 2013 Nets of Polyhedra The Problem of D urer, Shephard, et al Conjecture Every convex polyhedron has a non-overlapping edge unfolding. Facts about
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The Problem of D¨ urer, Shephard, et al
Conjecture
Every convex polyhedron has a non-overlapping edge unfolding.
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Facts about Edge Unfoldings
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Facts about Edge Unfoldings
◮ Every edge unfolding of a polyhedron is obtained by cutting
along a spanning tree of its edge skeleton.
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Facts about Edge Unfoldings
◮ Every edge unfolding of a polyhedron is obtained by cutting
along a spanning tree of its edge skeleton.
◮ Every edge unfolding yields an outerplanar graph with no cut
vertices.
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Facts about Edge Unfoldings
◮ Every edge unfolding of a polyhedron is obtained by cutting
along a spanning tree of its edge skeleton.
◮ Every edge unfolding yields an outerplanar graph with no cut
vertices.
◮ If G is an unfolding, then |V(G)| is even.
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Facts about Edge Unfoldings
◮ Every edge unfolding of a polyhedron is obtained by cutting
along a spanning tree of its edge skeleton.
◮ Every edge unfolding yields an outerplanar graph with no cut
vertices.
◮ If G is an unfolding, then |V(G)| is even. ◮ If G is an unfolding with |V(G)| = 2n, then no vertex has
degree exceeding n + 1.
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Facts about Edge Unfoldings
◮ Every edge unfolding of a polyhedron is obtained by cutting
along a spanning tree of its edge skeleton.
◮ Every edge unfolding yields an outerplanar graph with no cut
vertices.
◮ If G is an unfolding, then |V(G)| is even. ◮ If G is an unfolding with |V(G)| = 2n, then no vertex has
degree exceeding n + 1. Notation: ∆(G) is the maximum degree of G.
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Example
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The Main Problem
Characterize outerplanar graphs which have at least one polyhedral gluing.
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Polyhedral Gluings
Theorem (Steinitz)
A graph is the edge skeleton of a convex polyhedon iff it is simple, planar, and 3-connected.
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Polyhedral Gluings
Theorem (Steinitz)
A graph is the edge skeleton of a convex polyhedon iff it is simple, planar, and 3-connected.
Definition
A graph is “polyhedral” if it is simple, planar, and 3-connected.
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Spherical Gluings
Proposition
If G is an outerplanar graph with |V(G)| = 2n and no cut vertices, then G has precisely Cn =
(2n)!
n!(n + 1)! spherical gluings.
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Spherical Gluings
Proposition
If G is an outerplanar graph with |V(G)| = 2n and no cut vertices, then G has precisely Cn =
(2n)!
n!(n + 1)! spherical gluings. Draw a picture.
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The Main Problem (restated)
Find the outerplanar graphs for which the set of spherical gluings contains at least one polyhedral gluing.
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Does it glue?
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Does it glue?
Both do.
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Does it glue?
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Does it glue?
Yes.
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Does it glue?
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Does it glue?
No.
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Does it glue?
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Does it glue?
Yes.
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Does it glue?
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Does it glue?
No.
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The most we can say....
Conjecture
Suppose G is a maximal outerplanar graph with |V(G)| = 2n. Then G admits a polyhedral gluing iff ∆(G) ≤ n + 1.
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Proof by Induction?
Glue adjacent outer edges, then contract an edge. (Use the chalkboard.)
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Easy case
|V(G)| = 2n, ∆(G) = n + 1.
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Easy case
G has a “nice” degree-4 vertex.
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Easy case
G has a degree-5 vertex below a (2,3) flap.
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The Obstacle to Induction
A polyhedral gluing of a smaller graph may not extend to a polyhedral gluing of the larger graph. (Try it out!)
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The hard case
Most look like this....
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Another Problem
Characterize outerplanar graphs which have at most one polyhedral gluing.
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