Z 2 -embeddings and Tournaments Radoslav Fulek , Jan Kyn cl Z 2 - PowerPoint PPT Presentation

Z 2 -embeddings and Tournaments Radoslav Fulek , Jan Kynˇ cl

Z 2 -embeddings and Tournaments Radoslav Fulek , Jan Kynˇ cl June 12, 2018

Drawings of Graphs

Drawings of Graphs G = ( V, E ) is a simple graph . The set of vertices V is finite � V � and the set of edges E ⊆ . We treat G as a 1-dimensional 2 simplicial complex.

Drawings of Graphs G = ( V, E ) is a simple graph . The set of vertices V is finite � V � and the set of edges E ⊆ . We treat G as a 1-dimensional 2 simplicial complex. A drawing D of G on a 2-dimensional surface S is a generic and “ nice ” continuous map D : G → S . By “generic” we mean that the set of its self-intersections is finite and consisting only of transversal edge intersections, i.e., proper edge crossings .

Drawings of Graphs G = ( V, E ) is a simple graph . The set of vertices V is finite � V � and the set of edges E ⊆ . We treat G as a 1-dimensional 2 simplicial complex. A drawing D of G on a 2-dimensional surface S is a generic and “ nice ” continuous map D : G → S . By “generic” we mean that the set of its self-intersections is finite and consisting only of transversal edge intersections, i.e., proper edge crossings . Formally, D ( e ) is injective for every edge, C = { p ∈ S : | D − 1 [ p ] | > 1 } is finite, and every p ∈ C is a proper edge crossing of exactly two edges .

Drawings of Graphs G = ( V, E ) is a simple graph . The set of vertices V is finite � V � and the set of edges E ⊆ . We treat G as a 1-dimensional 2 simplicial complex. A drawing D of G on a 2-dimensional surface S is a generic and “ nice ” continuous map D : G → S . By “generic” we mean that the set of its self-intersections is finite and consisting only of transversal edge intersections, i.e., proper edge crossings .

Drawings of Graphs G = ( V, E ) is a simple graph . The set of vertices V is finite � V � and the set of edges E ⊆ . We treat G as a 1-dimensional 2 simplicial complex. A drawing D of G on a 2-dimensional surface S is a generic and “ nice ” continuous map D : G → S . By “generic” we mean that the set of its self-intersections is finite and consisting only of transversal edge intersections, i.e., proper edge crossings .

Drawings of Graphs G = ( V, E ) is a simple graph . The set of vertices V is finite � V � and the set of edges E ⊆ . We treat G as a 1-dimensional 2 simplicial complex. A drawing D of G on a 2-dimensional surface S is a generic and “ nice ” continuous map D : G → S . By “generic” we mean that the set of its self-intersections is finite and consisting only of transversal edge intersections, i.e., proper edge crossings .

Drawings of Graphs G = ( V, E ) is a simple graph . The set of vertices V is finite � V � and the set of edges E ⊆ . We treat G as a 1-dimensional 2 simplicial complex. A drawing D of G on a 2-dimensional surface S is a generic and “ nice ” continuous map D : G → S . By “generic” we mean that the set of its self-intersections is finite and consisting only of transversal edge intersections, i.e., proper edge crossings . Injective D is an embedding .

Z 2 -embeddings

Z 2 -embeddings Let D be a drawing of a graph G .

Z 2 -embeddings Let D be a drawing of a graph G . � E � Let I D ( G ) = {{ e, f } ∈ | e ∩ f = ∅ & | D ( e ) ∩ D ( f ) | = 2 1 } . 2 A drawing for which I D ( G ) = ∅ is a Z 2 -embedding .

Z 2 -embeddings Let D be a drawing of a graph G . � E � Let I D ( G ) = {{ e, f } ∈ | e ∩ f = ∅ & | D ( e ) ∩ D ( f ) | = 2 1 } . 2 A drawing for which I D ( G ) = ∅ is a Z 2 -embedding . Theorem 1 (Hanani–Tutte, 1934–1970) . If G admits a Z 2 -embedding in the plane then G is planar.

Z 2 -embeddings Let D be a drawing of a graph G . � E � Let I D ( G ) = {{ e, f } ∈ | e ∩ f = ∅ & | D ( e ) ∩ D ( f ) | = 2 1 } . 2 A drawing for which I D ( G ) = ∅ is a Z 2 -embedding . Theorem 1 (Hanani–Tutte, 1934–1970) . If G admits a Z 2 -embedding in the plane then G is planar. � E Let I ◦ � D ( G ) = {{ e, f } ∈ | | D ( e ) ∩ D ( f ) | = 2 1 } . 2 A drawing for which I ◦ D ( G ) = ∅ is a strong Z 2 -embedding .

Z 2 -embeddings Let D be a drawing of a graph G . � E � Let I D ( G ) = {{ e, f } ∈ | e ∩ f = ∅ & | D ( e ) ∩ D ( f ) | = 2 1 } . 2 A drawing for which I D ( G ) = ∅ is a Z 2 -embedding . Theorem 1 (Hanani–Tutte, 1934–1970) . If G admits a Z 2 -embedding in the plane then G is planar. � E Let I ◦ � D ( G ) = {{ e, f } ∈ | | D ( e ) ∩ D ( f ) | = 2 1 } . 2 A drawing for which I ◦ D ( G ) = ∅ is a strong Z 2 -embedding . Theorem 2 (Cairns and Nikolayevsky 2000, Pelsmajer, Schaefer, and ˇ Stefankoviˇ c 2009) . If a graph G admits a strong Z 2 -embedding on S then G can be embedded on S .

Z 2 -rotation Order Type

Z 2 -rotation Order Type Let D be a drawing of G = ( V, E ) on a surface S .

Z 2 -rotation Order Type Let D be a drawing of G = ( V, E ) on a surface S . For v ∈ e, f, g ∈ E , o D ( e, f, g ) = +1 and o D ( e, f, g ) = − 1 if e, f and g appear ccw and cw, resp., in the rotation at v . e e o D ( e, f, g ) = − 1 o D ( e, f, g ) = +1 v v g f f g

Z 2 -rotation Order Type Let D be a drawing of G = ( V, E ) on a surface S . For v ∈ e, f, g ∈ E , o D ( e, f, g ) = +1 and o D ( e, f, g ) = − 1 if e, f and g appear ccw and cw, resp., in the rotation at v . e e o D ( e, f, g ) = − 1 o D ( e, f, g ) = +1 v v g f f g σ D ( e, f, g ) = o D ( e, f, g ) · ( − 1) cr( { e,f,g } ) , where cr( { e, f, g } ) = | D ( e ) ∩ D ( f ) | + | D ( e ) ∩ D ( g ) | + | D ( f ) ∩ D ( g ) |

Z 2 -rotation Order Type Let D be a drawing of G = ( V, E ) on a surface S . For v ∈ e, f, g ∈ E , o D ( e, f, g ) = +1 and o D ( e, f, g ) = − 1 if e, f and g appear ccw and cw, resp., in the rotation at v . e e o D ( e, f, g ) = − 1 o D ( e, f, g ) = +1 v v g f f g σ D ( e, f, g ) = o D ( e, f, g ) · ( − 1) cr( { e,f,g } ) , where cr( { e, f, g } ) = | D ( e ) ∩ D ( f ) | + | D ( e ) ∩ D ( g ) | + | D ( f ) ∩ D ( g ) | e e σ D ( e, f, g ) does v not change after a v g f g f flip

Z 2 -rotation Order Type (cont’) For v ∈ e, f, g ∈ E , o D ( e, f, g ) = +1 and o D ( e, f, g ) = − 1 if e, f and g appear ccw and cw, resp., in the rotation at v . σ D ( e, f, g ) = o D ( e, f, g ) · ( − 1) cr( { e,f,g } ) , where cr( { e, f, g } ) = | D ( e ) ∩ D ( f ) | + | D ( e ) ∩ D ( g ) | + | D ( f ) ∩ D ( g ) |

Z 2 -rotation Order Type (cont’) For v ∈ e, f, g ∈ E , o D ( e, f, g ) = +1 and o D ( e, f, g ) = − 1 if e, f and g appear ccw and cw, resp., in the rotation at v . σ D ( e, f, g ) = o D ( e, f, g ) · ( − 1) cr( { e,f,g } ) , where cr( { e, f, g } ) = | D ( e ) ∩ D ( f ) | + | D ( e ) ∩ D ( g ) | + | D ( f ) ∩ D ( g ) | We count the number of 3 element subsets of { e, f, g, h ∋ v } for which σ D and o D return the same value.

Z 2 -rotation Order Type (cont’) For v ∈ e, f, g ∈ E , o D ( e, f, g ) = +1 and o D ( e, f, g ) = − 1 if e, f and g appear ccw and cw, resp., in the rotation at v . σ D ( e, f, g ) = o D ( e, f, g ) · ( − 1) cr( { e,f,g } ) , where cr( { e, f, g } ) = | D ( e ) ∩ D ( f ) | + | D ( e ) ∩ D ( g ) | + | D ( f ) ∩ D ( g ) | We count the number of 3 element subsets of { e, f, g, h ∋ v } for which σ D and o D return the same value. Claim 1. |{{ e 1 , e 2 , e 3 } ⊂ { e, f, g, h } : σ D ( e 1 , e 2 , e 3 ) = o D ( e 1 , e 2 , e 3 ) }| = 2 0

Z 2 -rotation Order Type (cont’) For v ∈ e, f, g ∈ E , o D ( e, f, g ) = +1 and o D ( e, f, g ) = − 1 if e, f and g appear ccw and cw, resp., in the rotation at v . σ D ( e, f, g ) = o D ( e, f, g ) · ( − 1) cr( { e,f,g } ) , where cr( { e, f, g } ) = | D ( e ) ∩ D ( f ) | + | D ( e ) ∩ D ( g ) | + | D ( f ) ∩ D ( g ) | We count the number of 3 element subsets of { e, f, g, h ∋ v } for which σ D and o D return the same value. Claim 1. |{{ e 1 , e 2 , e 3 } ⊂ { e, f, g, h } : σ D ( e 1 , e 2 , e 3 ) = o D ( e 1 , e 2 , e 3 ) }| = 2 0 Proof. We count the number of 3 element subsets for which cr( { e 1 , e 2 , e 3 } ) = 2 0 . Thus, we count the number of triples of vertices in a graph with 4 vertices inducing an even number of edges. This number must be even.

Z 2 -rotation Order Type (cont’)

Z 2 -rotation Order Type (cont’) ♣ D ( v ) := {{ e, f, g } : v ∈ e, f, g and σ D ( e, f, g ) = o D ( e, f, g ) }

Z 2 -rotation Order Type (cont’) ♣ D ( v ) := {{ e, f, g } : v ∈ e, f, g and σ D ( e, f, g ) = o D ( e, f, g ) } � δ ( v ) � Claim 2. Let D be a Z 2 -embedding. If ♣ D ( v ) = , for all 3 v ∈ V , then D can be made strong while keeping the rotation at every vertex.

Z 2 -rotation Order Type (cont’) ♣ D ( v ) := {{ e, f, g } : v ∈ e, f, g and σ D ( e, f, g ) = o D ( e, f, g ) } � δ ( v ) � Claim 2. Let D be a Z 2 -embedding. If ♣ D ( v ) = , for all 3 v ∈ V , then D can be made strong while keeping the rotation at every vertex. Proof. Let G aux ( v ) = ( δ ( v ) , E ′ ) , where ef ∈ E ′ , if | D ( e ) ∩ D ( f ) | = 2 1 . G aux ( v ) must be a complete bipartite graph. Pushing every edge in one part over v renders G aux ( v ) empty.