Recursive regularity and the finest regular coarsening of a - PowerPoint PPT Presentation

Proposition Recursively regular subdivisions are a superset of the regular subdivisions and a subset of the acyclic ones Proof Assume that it exists a cycle in a recursively regular subdivision. We can assume that the cycle belongs to more than one cell of the FRC and apply recursion otherwise. Then, the FRC cannot be regular, since a cycle cannot be destroyed by means of merging cells.

Proposition There is a 5-dimensional point set with at least 12 recursively regular triangulations pairwise disconnected and disconnected from any regular triangulation in the graph of flips

Applications Floodlights Homothecies Graph embeddings Spider webs

Floodlights polyhedral fan with n cells

Floodlights n points polyhedral fan with n cells

Floodlights n points one to one polyhedral fan with n cells

Floodlights n points one to one polyhedral fan with n cells covering assignment

Covering fans A polyhedral fan is universally covering if for any point set there exists a covering assignment

Covering fans A polyhedral fan is universally covering if for any point set there exists a covering assignment Theorem (Galperin & Galperin, 1981) Regular fans are universally covering

Covering fans A polyhedral fan is universally covering if for any point set there exists a covering assignment Theorem (Galperin & Galperin, 1981) Regular fans are universally covering Theorem Recursively regular fans are universally covering Cyclic fans are not universally covering

Fans Cyclic Rec. regular Regular

Fans Cyclic Rec. regular Regular Universally covering

Fans Cyclic Rec. regular Regular Universally covering

Fans Cyclic Rec. regular Regular Universally covering

Fans Cyclic Rec. regular Regular Universally covering

Overlapping condition

Overlapping condition

Overlapping condition

Directional graph embeddings → − V = { 1 , ..., n } E ⊂ V × V

Directional graph embeddings → − V = { 1 , ..., n } E ⊂ V × V P = { p 1 , ...p n } ⊂ R d

Directional graph embeddings → − N : − → → R d V = { 1 , ..., n } E ⊂ V × V E − P = { p 1 , ...p n } ⊂ R d

Directional graph embeddings → − N : − → → R d V = { 1 , ..., n } E ⊂ V × V E − P = { p 1 , ...p n } ⊂ R d proper embedding Is there a σ : V ↔ P N ( i, j ) · p i ≥ N ( i, j ) · p j ∀ ( i, j ) ∈ − → E ?

Directional graph embeddings → − N : − → → R d V = { 1 , ..., n } E ⊂ V × V E − P = { p 1 , ...p n } ⊂ R d proper embedding Is there a σ : V ↔ P N ( i, j ) · p i ≥ N ( i, j ) · p j ∀ ( i, j ) ∈ − → E ? A path is always embeddable

Directional graph embeddings → − N : − → → R d V = { 1 , ..., n } E ⊂ V × V E − P = { p 1 , ...p n } ⊂ R d proper embedding Is there a σ : V ↔ P N ( i, j ) · p i ≥ N ( i, j ) · p j ∀ ( i, j ) ∈ − → E ? A path is always embeddable A cycle is not always embeddable

Directional graph embeddings → − N : − → → R d V = { 1 , ..., n } E ⊂ V × V E − P = { p 1 , ...p n } ⊂ R d proper embedding Is there a σ : V ↔ P N ( i, j ) · p i ≥ N ( i, j ) · p j ∀ ( i, j ) ∈ − → E ? A path is always embeddable A cycle is not always embeddable A tree?

Proposition If the graph is the normal graph (or a projection of a subgraph) of a fan and there is a covering assignment for P , there is a proper graph embedding for P . If there is no assignment satisfying OC for... , there is no proper embedding

Proposition If the graph is the normal graph (or a projection of a subgraph) of a fan and there is a covering assignment for P , there is a proper graph embedding for P . If there is no assignment satisfying OC for... , there is no proper embedding Corollary The graph of a polytope is always embeddable.

Homothecies 1,3 1,1 1,2 0,6 0,9 1,5

Homothecies 1,3 1,1 1,2 0,6 0,9 1,5

Homothecies

Homothecies

Homothecies

Homothecies