Preliminaries Results Approaching the Problem Remarks and Open Problems Three Edge Lengths Suffice For Drawing Outerplanar Graphs Noga Alon Ohad N. Feldheim Tel-Aviv University IMU, 2012 Noga Alon and Ohad N. Feldheim Three Edge Lengths Suffice For Drawing Outerplanar Graphs
Preliminaries Embedding in the Plane Results Drawings Approaching the Problem Degenerate Drawings Remarks and Open Problems Mapping Graphs to the Plane Let G = ( V G , E G ) , π : V G → C , v 0 v 1 v 2 v 0 v 2 → v 1 v 3 v 4 , v 3 v 4 π ( G ) Graph G Noga Alon and Ohad N. Feldheim Three Edge Lengths Suffice For Drawing Outerplanar Graphs
Preliminaries Embedding in the Plane Results Drawings Approaching the Problem Degenerate Drawings Remarks and Open Problems Mapping Graphs to the Plane Let G = ( V G , E G ) , π : V G → C , e = ( v 0 , v 1 ). We set: π ( e ) := ( π ( v 0 ) , π ( v 1 )) v 0 v 1 v 2 v 0 v 2 → v 1 v 3 v 4 , v 3 v 4 π ( G ) Graph G Noga Alon and Ohad N. Feldheim Three Edge Lengths Suffice For Drawing Outerplanar Graphs
Preliminaries Embedding in the Plane Results Drawings Approaching the Problem Degenerate Drawings Remarks and Open Problems Mapping Graphs to the Plane Let G = ( V G , E G ) , π : V G → C , e = ( v 0 , v 1 ). We set: π ( e ) := ( π ( v 0 ) , π ( v 1 )) len π ( e ) := | π ( v 0 ) − π ( v 1 ) | v 0 v 1 v 2 v 0 v 2 → v 1 v 3 v 4 , v 3 v 4 π ( G ) Graph G Noga Alon and Ohad N. Feldheim Three Edge Lengths Suffice For Drawing Outerplanar Graphs
Preliminaries Embedding in the Plane Results Drawings Approaching the Problem Degenerate Drawings Remarks and Open Problems Mapping Graphs to the Plane Let G = ( V G , E G ) , π : V G → C , e = ( v 0 , v 1 ). We set: π ( e ) := ( π ( v 0 ) , π ( v 1 )) len π ( e ) := | π ( v 0 ) − π ( v 1 ) | lens( π ) := |{ len π ( e ) : e ∈ E G }| v 0 v 1 v 2 v 0 v 2 → v 1 v 3 v 4 , v 3 v 4 π ( G ) Graph G Noga Alon and Ohad N. Feldheim Three Edge Lengths Suffice For Drawing Outerplanar Graphs
Preliminaries Embedding in the Plane Results Drawings Approaching the Problem Degenerate Drawings Remarks and Open Problems Drawings Let G = ( V G , E G ) , π : V G → C , Drawing : ∀ v , v ′ ∈ V G , ∀ e ∈ E G : π ( v ) � = π ( v ′ ), π ( v ) / ∈ π ( e ) v 1 v 1 v 0 v 0 v 2 v 2 → v 3 v 3 v 4 v 4 π ( G ) Graph G Noga Alon and Ohad N. Feldheim Three Edge Lengths Suffice For Drawing Outerplanar Graphs
Preliminaries Embedding in the Plane Results Drawings Approaching the Problem Degenerate Drawings Remarks and Open Problems Drawings Let G = ( V G , E G ) , π : V G → C , Drawing : ∀ v , v ′ ∈ V G , ∀ e ∈ E G : π ( v ) � = π ( v ′ ), π ( v ) / ∈ π ( e ) Distance Number dn( G ) := min { lens( π ) : π is a drawing of G } v 1 v 1 v 0 v 0 v 2 v 2 → v 3 v 3 v 4 v 4 π ( G ) Graph G Noga Alon and Ohad N. Feldheim Three Edge Lengths Suffice For Drawing Outerplanar Graphs
Preliminaries Embedding in the Plane Results Drawings Approaching the Problem Degenerate Drawings Remarks and Open Problems Degenerate Drawings Let G = ( V G , E G ) , π : V G → C , Deg. Drawing : ∀ v , v ′ ∈ V G : π ( v ) � = π ( v ′ ) v 1 v 0 v 1 v 0 v 2 v 2 → v 3 v 3 v 4 v 4 π ( G ) Graph G Noga Alon and Ohad N. Feldheim Three Edge Lengths Suffice For Drawing Outerplanar Graphs
Preliminaries Embedding in the Plane Results Drawings Approaching the Problem Degenerate Drawings Remarks and Open Problems Degenerate Drawings Let G = ( V G , E G ) , π : V G → C , Deg. Drawing : ∀ v , v ′ ∈ V G : π ( v ) � = π ( v ′ ) Degenerate Distance Number ddn( G ) := min { lens( π ) : π is a deg. drawing of G } v 1 v 0 v 1 v 0 v 2 v 2 → v 3 v 3 v 4 v 4 π ( G ) Graph G Noga Alon and Ohad N. Feldheim Three Edge Lengths Suffice For Drawing Outerplanar Graphs
Preliminaries Embedding in the Plane Results Drawings Approaching the Problem Degenerate Drawings Remarks and Open Problems Properties of the Distance Number Properties of dn and ddn Let G ⊂ H . dn( G ) ≤ dn( H ) ddn( G ) ≤ ddn( H ) ddn( G ) ≤ dn( G ) Noga Alon and Ohad N. Feldheim Three Edge Lengths Suffice For Drawing Outerplanar Graphs
Preliminaries Results Previous Results Approaching the Problem Our Results Remarks and Open Problems Distance Number - Previous Results What bounds can we get on dn, ddn? Are they ever different? Noga Alon and Ohad N. Feldheim Three Edge Lengths Suffice For Drawing Outerplanar Graphs
Preliminaries Results Previous Results Approaching the Problem Our Results Remarks and Open Problems Distance Number - Previous Results What bounds can we get on dn, ddn? Are they ever different? Results on K n c 1 n c 2 n Guth-Katz(’11)/Erd˝ os(’46): log n ≤ ddn( K n ) ≤ √ log n Noga Alon and Ohad N. Feldheim Three Edge Lengths Suffice For Drawing Outerplanar Graphs
Preliminaries Results Previous Results Approaching the Problem Our Results Remarks and Open Problems Distance Number - Previous Results What bounds can we get on dn, ddn? Are they ever different? Results on K n c 1 n c 2 n Guth-Katz(’11)/Erd˝ os(’46): log n ≤ ddn( K n ) ≤ √ log n n − 1 ≤ dn( K n ) ≤ n Szemer´ edi(’95)/Erd˝ os(’51): 3 2 Noga Alon and Ohad N. Feldheim Three Edge Lengths Suffice For Drawing Outerplanar Graphs
Preliminaries Results Previous Results Approaching the Problem Our Results Remarks and Open Problems Distance Number - Previous Results What bounds can we get on dn, ddn? Are they ever different? Results on K n c 1 n c 2 n Guth-Katz(’11)/Erd˝ os(’46): log n ≤ ddn( K n ) ≤ √ log n n − 1 ≤ dn( K n ) ≤ n Szemer´ edi(’95)/Erd˝ os(’51): 3 2 Results on graphs with bdd. degree ∆, n vertices Carmi, Dujmovi´ c, Morin, Wood(’08): For ∆ ≥ 5: ddn is not uniformly bdd. For ∆ ≥ 7: exist G n with ddn( G ) = Ω( n c ) for c (∆) < C < 1. dn( G ) = O (∆ 4 log n ) if G n ’s treewidth is uniformly bdd. Noga Alon and Ohad N. Feldheim Three Edge Lengths Suffice For Drawing Outerplanar Graphs
Preliminaries Results Previous Results Approaching the Problem Our Results Remarks and Open Problems Distance Number - Planar Graphs Planar Graph: Has a drawing without crossings. Noga Alon and Ohad N. Feldheim Three Edge Lengths Suffice For Drawing Outerplanar Graphs
Preliminaries Results Previous Results Approaching the Problem Our Results Remarks and Open Problems Distance Number - Planar Graphs Planar Graph: Has a drawing without crossings. ∃ planar G n with ddn( G ) = Ω( √ n ). Noga Alon and Ohad N. Feldheim Three Edge Lengths Suffice For Drawing Outerplanar Graphs
Preliminaries Results Previous Results Approaching the Problem Our Results Remarks and Open Problems Distance Number - Planar Graphs Planar Graph: Has a drawing without crossings. ∃ planar G n with ddn( G ) = Ω( √ n ). Carmi, Dujmovic, Morin, Wood(’08): Do outerplanar graphs have uniformly bounded (degenerate) distance number? Noga Alon and Ohad N. Feldheim Three Edge Lengths Suffice For Drawing Outerplanar Graphs
Preliminaries Results Previous Results Approaching the Problem Our Results Remarks and Open Problems Outerplanar Graphs Outerplanar Graph: ∃ drawing, without crossings, s.t. unbounded face contains all vertices. Outerplanar graph Noga Alon and Ohad N. Feldheim Three Edge Lengths Suffice For Drawing Outerplanar Graphs
Preliminaries Results Previous Results Approaching the Problem Our Results Remarks and Open Problems Outerplanar Graphs Outerplanar Graph: ∃ drawing, without crossings, s.t. unbounded face contains all vertices. Triangulated Outerplanar Graph: Outerplanar graph where all bounded faces are triangles. Trinagulated outerplanar Outerplanar graph graph Noga Alon and Ohad N. Feldheim Three Edge Lengths Suffice For Drawing Outerplanar Graphs
Preliminaries Results Previous Results Approaching the Problem Our Results Remarks and Open Problems Our Results Carmi, Dujmovic, Morin, Wood(’08): Do outerplanar graphs have uniformly bounded (degenerate) distance number? Noga Alon and Ohad N. Feldheim Three Edge Lengths Suffice For Drawing Outerplanar Graphs
Preliminaries Results Previous Results Approaching the Problem Our Results Remarks and Open Problems Our Results Carmi, Dujmovic, Morin, Wood(’08): Do outerplanar graphs have uniformly bounded (degenerate) distance number? Theorem (Alon, F.) For almost every triple a , b , c ∈ (0 , 1), every outerplanar graph has a degenerate drawing using only edge-lengths a , b and c . Noga Alon and Ohad N. Feldheim Three Edge Lengths Suffice For Drawing Outerplanar Graphs
Preliminaries Results Previous Results Approaching the Problem Our Results Remarks and Open Problems Our Results Carmi, Dujmovic, Morin, Wood(’08): Do outerplanar graphs have uniformly bounded (degenerate) distance number? Theorem (Alon, F.) For almost every triple a , b , c ∈ (0 , 1), every outerplanar graph has a degenerate drawing using only edge-lengths a , b and c . Work in progress... For almost every nine values a 0 , ..., a 8 ∈ (0 , 1), every outerplanar graph has a drawing using only edge-lengths a 0 , ..., a 8 . Noga Alon and Ohad N. Feldheim Three Edge Lengths Suffice For Drawing Outerplanar Graphs
Drawing Outerplanar Graphs
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