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

Z2-embeddings and Tournaments

Radoslav Fulek, Jan Kynˇ cl

Z2-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 and the set of edges E ⊆ V

2

• . We treat G as a 1-dimensional

simplicial complex.

Drawings of Graphs G = (V, E) is a simple graph. The set of vertices V is finite and the set of edges E ⊆ V

2

• . We treat G as a 1-dimensional

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

• f 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 and the set of edges E ⊆ V

2

• . We treat G as a 1-dimensional

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

• f 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 and the set of edges E ⊆ V

2

• . We treat G as a 1-dimensional

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

• f 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 and the set of edges E ⊆ V

2

• . We treat G as a 1-dimensional

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

• f 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 and the set of edges E ⊆ V

2

• . We treat G as a 1-dimensional

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

• f 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 and the set of edges E ⊆ V

2

• . We treat G as a 1-dimensional

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

• f transversal edge intersections, i.e., proper edge crossings.

Injective D is an embedding.

Z2-embeddings

Z2-embeddings Let D be a drawing of a graph G.

Z2-embeddings Let D be a drawing of a graph G. Let ID(G) = {{e, f} ∈ E

2

• | e ∩ f = ∅ & |D(e) ∩ D(f)| =2 1}.

A drawing for which ID(G) = ∅ is a Z2-embedding.

Z2-embeddings Let D be a drawing of a graph G. Let ID(G) = {{e, f} ∈ E

2

• | e ∩ f = ∅ & |D(e) ∩ D(f)| =2 1}.

A drawing for which ID(G) = ∅ is a Z2-embedding. Theorem 1 (Hanani–Tutte, 1934–1970). If G admits a Z2-embedding in the plane then G is planar.

Z2-embeddings Let D be a drawing of a graph G. Let ID(G) = {{e, f} ∈ E

2

• | e ∩ f = ∅ & |D(e) ∩ D(f)| =2 1}.

A drawing for which ID(G) = ∅ is a Z2-embedding. Let I◦

D(G) = {{e, f} ∈

E

2

• | |D(e) ∩ D(f)| =2 1}.

A drawing for which I◦

D(G) = ∅ is a strong Z2-embedding.

Theorem 1 (Hanani–Tutte, 1934–1970). If G admits a Z2-embedding in the plane then G is planar.

Z2-embeddings Let D be a drawing of a graph G. Let ID(G) = {{e, f} ∈ E

2

• | e ∩ f = ∅ & |D(e) ∩ D(f)| =2 1}.

A drawing for which ID(G) = ∅ is a Z2-embedding. Let I◦

D(G) = {{e, f} ∈

E

2

• | |D(e) ∩ D(f)| =2 1}.

A drawing for which I◦

D(G) = ∅ is a strong Z2-embedding.

Theorem 1 (Hanani–Tutte, 1934–1970). If G admits a Z2-embedding in the plane then G is planar. Theorem 2 (Cairns and Nikolayevsky 2000, Pelsmajer, Schaefer, and ˇ Stefankoviˇ c 2009). If a graph G admits a strong Z2-embedding on S then G can be embedded on S.

Z2-rotation Order Type

Z2-rotation Order Type Let D be a drawing of G = (V, E) on a surface S.

Z2-rotation Order Type Let D be a drawing of G = (V, E) on a surface S. For v ∈ e, f, g ∈ E, oD(e, f, g) = +1 and oD(e, f, g) = −1 if e, f and g appear ccw and cw, resp., in the rotation at v. v e f g v e g f

• D(e, f, g) = +1
• D(e, f, g) = −1

Z2-rotation Order Type Let D be a drawing of G = (V, E) on a surface S. For v ∈ e, f, g ∈ E, oD(e, f, g) = +1 and oD(e, f, g) = −1 if e, f and g appear ccw and cw, resp., in the rotation at v. σD(e, f, g) = oD(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)| v e f g v e g f

• D(e, f, g) = +1
• D(e, f, g) = −1

Z2-rotation Order Type Let D be a drawing of G = (V, E) on a surface S. For v ∈ e, f, g ∈ E, oD(e, f, g) = +1 and oD(e, f, g) = −1 if e, f and g appear ccw and cw, resp., in the rotation at v. σD(e, f, g) = oD(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)| v e f g v e g f

• D(e, f, g) = +1
• D(e, f, g) = −1

σD(e, f, g) does not change after a flip e f g v g f e v

Z2-rotation Order Type (cont’) For v ∈ e, f, g ∈ E, oD(e, f, g) = +1 and oD(e, f, g) = −1 if e, f and g appear ccw and cw, resp., in the rotation at v. σD(e, f, g) = oD(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)|

Z2-rotation Order Type (cont’) For v ∈ e, f, g ∈ E, oD(e, f, g) = +1 and oD(e, f, g) = −1 if e, f and g appear ccw and cw, resp., in the rotation at v. σD(e, f, g) = oD(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 σD and oD return the same value.

Z2-rotation Order Type (cont’) For v ∈ e, f, g ∈ E, oD(e, f, g) = +1 and oD(e, f, g) = −1 if e, f and g appear ccw and cw, resp., in the rotation at v. σD(e, f, g) = oD(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)|

Claim 1. |{{e1, e2, e3} ⊂ {e, f, g, h} : σD(e1, e2, e3) = oD(e1, e2, e3)}| =2 0

We count the number of 3 element subsets of {e, f, g, h ∋ v} for which σD and oD return the same value.

Z2-rotation Order Type (cont’) For v ∈ e, f, g ∈ E, oD(e, f, g) = +1 and oD(e, f, g) = −1 if e, f and g appear ccw and cw, resp., in the rotation at v. σD(e, f, g) = oD(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)|

Claim 1. |{{e1, e2, e3} ⊂ {e, f, g, h} : σD(e1, e2, e3) = oD(e1, e2, e3)}| =2 0

We count the number of 3 element subsets of {e, f, g, h ∋ v} for which σD and oD return the same value.

• Proof. We count the number of 3 element subsets for which

cr({e1, e2, e3}) =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.

Z2-rotation Order Type (cont’)

Z2-rotation Order Type (cont’) ♣D(v) := {{e, f, g} : v ∈ e, f, g and σD(e, f, g) = oD(e, f, g)}

Z2-rotation Order Type (cont’) ♣D(v) := {{e, f, g} : v ∈ e, f, g and σD(e, f, g) = oD(e, f, g)} Claim 2. Let D be a Z2-embedding. If ♣D(v) = δ(v)

3

• , for all

v ∈ V , then D can be made strong while keeping the rotation at every vertex.

Z2-rotation Order Type (cont’) ♣D(v) := {{e, f, g} : v ∈ e, f, g and σD(e, f, g) = oD(e, f, g)} Claim 2. Let D be a Z2-embedding. If ♣D(v) = δ(v)

3

• , for all

v ∈ V , then D can be made strong while keeping the rotation at every vertex.

• Proof. Let Gaux(v) = (δ(v), E′), where ef ∈ E′, if

|D(e) ∩ D(f)| =2 1. Gaux(v) must be a complete bipartite

• graph. Pushing every edge in one part over v renders Gaux(v)

empty.

Z2-rotation Order Type (cont’)

Claim 1. |{{e1, e2, e3} ⊂ {e, f, g, h} : σD(e1, e2, e3) = oD(e1, e2, e3)}| =2 0

♣D(v) := {{e, f, g} : v ∈ e, f, g and σD(e, f, g) = oD(e, f, g)} Claim 2. Let D be a Z2-embedding. If ♣D(v) = δ(v)

3

• , for all

v ∈ V , then D can be made strong while keeping the rotation at every vertex.

Z2-rotation Order Type (cont’)

Claim 1. |{{e1, e2, e3} ⊂ {e, f, g, h} : σD(e1, e2, e3) = oD(e1, e2, e3)}| =2 0

♣D(v) := {{e, f, g} : v ∈ e, f, g and σD(e, f, g) = oD(e, f, g)}

Corollary 1. Let e ∈ E such that v ∈ e. ♣D(v) = ∆{f,g},oD(e,f,g)=σD(e,f,g){{e′, f, g} : σD(e′, f, g) = oD(e′, f, g)}

Claim 2. Let D be a Z2-embedding. If ♣D(v) = δ(v)

3

• , for all

v ∈ V , then D can be made strong while keeping the rotation at every vertex.

Z2-rotation Order Type (cont’)

Claim 1. |{{e1, e2, e3} ⊂ {e, f, g, h} : σD(e1, e2, e3) = oD(e1, e2, e3)}| =2 0

♣D(v) := {{e, f, g} : v ∈ e, f, g and σD(e, f, g) = oD(e, f, g)}

Corollary 1. Let e ∈ E such that v ∈ e. ♣D(v) = ∆{f,g},oD(e,f,g)=σD(e,f,g){{e′, f, g} : σD(e′, f, g) = oD(e′, f, g)}

• Proof. Obviously, {e, f, g} ∈ ♣D(v) iff it appears as an element of exactly
• ne summand of ∆. Let {e′, f, g}, e′ = e. Then by Claim 1. applied to

{e, e′, f, g}, {e′, f, g} ∈ ♣D(v) iff it appears once or three times.

Claim 2. Let D be a Z2-embedding. If ♣D(v) = δ(v)

3

• , for all

v ∈ V , then D can be made strong while keeping the rotation at every vertex.

Z2-rotation Tournaments

Z2-rotation Tournaments Let v ∈ e ∈ E. The Z2-rotation tournament is the tournament TD(v, e) on {f ∈ E : v ∈ f} \ {e} s.t. − → fg if σD(e, f, g) = +1.

Z2-rotation Tournaments Let v ∈ e ∈ E. The Z2-rotation tournament is the tournament TD(v, e) on {f ∈ E : v ∈ f} \ {e} s.t. − → fg if σD(e, f, g) = +1. Claim 3. For every pair e, f ∈ v ∈ E, TD(v, e) is acyclic if and

• nly if TD(v, f) is acyclic.

Z2-rotation Tournaments Let v ∈ e ∈ E. The Z2-rotation tournament is the tournament TD(v, e) on {f ∈ E : v ∈ f} \ {e} s.t. − → fg if σD(e, f, g) = +1. Claim 3. For every pair e, f ∈ v ∈ E, TD(v, e) is acyclic if and

• nly if TD(v, f) is acyclic.
• Proof. If TD(v, e) is acyclic then we obtain D′ so that

σD′(e, g, h) = oD′(e, g, h) for all g, h ∋ v. It follows by Claim 1. that σD′(e′, g, h) = oD′(e′, g, h) for all e′, g, h ∋ v. By the corollary, applied with e := f and D := D′ we obtain σD(f, g, h) = σD′(f, g, h) = oD′(f, g, h).

Z2-rotation Tournaments Let v ∈ e ∈ E. The Z2-rotation tournament is the tournament TD(v, e) on {f ∈ E : v ∈ f} \ {e} s.t. − → fg if σD(e, f, g) = +1. Claim 3. For every pair e, f ∈ v ∈ E, TD(v, e) is acyclic if and

• nly if TD(v, f) is acyclic.
• Proof. If TD(v, e) is acyclic then we obtain D′ so that

σD′(e, g, h) = oD′(e, g, h) for all g, h ∋ v. It follows by Claim 1. that σD′(e′, g, h) = oD′(e′, g, h) for all e′, g, h ∋ v. By the corollary, applied with e := f and D := D′ we obtain σD(f, g, h) = σD′(f, g, h) = oD′(f, g, h).

Z2-rotation Tournaments Let v ∈ e ∈ E. The Z2-rotation tournament is the tournament TD(v, e) on {f ∈ E : v ∈ f} \ {e} s.t. − → fg if σD(e, f, g) = +1. Claim 3. For every pair e, f ∈ v ∈ E, TD(v, e) is acyclic if and

• nly if TD(v, f) is acyclic.

A drawing D of a graph G is Z2-acyclic if TD(v, e) is acyclic for all v ∈ e ∈ E.

Z2-rotation Tournaments Let v ∈ e ∈ E. The Z2-rotation tournament is the tournament TD(v, e) on {f ∈ E : v ∈ f} \ {e} s.t. − → fg if σD(e, f, g) = +1. Claim 3. For every pair e, f ∈ v ∈ E, TD(v, e) is acyclic if and

• nly if TD(v, f) is acyclic.

A drawing D of a graph G is Z2-acyclic if TD(v, e) is acyclic for all v ∈ e ∈ E. Corollary 2. If G admits a Z2-acyclic Z2-embedding D on a surface S then G admits a strong Z2-embedding on S.

Making Z2-acyclic Z2-embeddings Strong

Making Z2-acyclic Z2-embeddings Strong Theorem 2 (Cairns and Nikolayevsky 2000, Pelsmajer, Schaefer, and ˇ Stefankoviˇ c 2009). If a graph G admits a strong Z2-embedding on S then G can be embedded on S.

Making Z2-acyclic Z2-embeddings Strong Theorem 2 (Cairns and Nikolayevsky 2000, Pelsmajer, Schaefer, and ˇ Stefankoviˇ c 2009). If a graph G admits a strong Z2-embedding on S then G can be embedded on S. Corollary 3. If G admits a Z2-acyclic Z2-embedding D on S then G can be embedded on S.

Making Z2-acyclic Z2-embeddings Strong Theorem 2 (Cairns and Nikolayevsky 2000, Pelsmajer, Schaefer, and ˇ Stefankoviˇ c 2009). If a graph G admits a strong Z2-embedding on S then G can be embedded on S. Corollary 3. If G admits a Z2-acyclic Z2-embedding D on S then G can be embedded on S. Corollary 4. If the restrictions of a drawing D of G to all 4-stars of G are Z2-acyclic then D is Z2-acyclic.

Making Z2-acyclic Z2-embeddings Strong Theorem 2 (Cairns and Nikolayevsky 2000, Pelsmajer, Schaefer, and ˇ Stefankoviˇ c 2009). If a graph G admits a strong Z2-embedding on S then G can be embedded on S. Corollary 3. If G admits a Z2-acyclic Z2-embedding D on S then G can be embedded on S. Claim 4. Every planar Z2-embedding of a 3-connected graph G is Z2-acyclic. Corollary 4. If the restrictions of a drawing D of G to all 4-stars of G are Z2-acyclic then D is Z2-acyclic.

K3,n = ({a, b, c} ∪ {0, . . . , n − 1}, {a0, . . . , a(n − 1), b0, c0}) K3,n

K3,n = ({a, b, c} ∪ {0, . . . , n − 1}, {a0, . . . , a(n − 1), b0, c0}) K3,n a c 1 2 3 . . . b a c 1 2 3 . . . b Let T be the spanning tree in K3,n with edges a0, b0, c0, a1, . . . , a(n − 1). T

K3,n = ({a, b, c} ∪ {0, . . . , n − 1}, {a0, . . . , a(n − 1), b0, c0}) K3,n a c 1 2 3 . . . b a c 1 2 3 . . . b Let T be the spanning tree in K3,n with edges a0, b0, c0, a1, . . . , a(n − 1). T Let D be a drawing of K3,n in the plane such that |D(e) ∩ D(f)|2 = 0 if e ∈ T and e ∩ f = ∅.

K3,n = ({a, b, c} ∪ {0, . . . , n − 1}, {a0, . . . , a(n − 1), b0, c0}) K3,n a c 1 2 3 . . . b a c 1 2 3 . . . b Let T be the spanning tree in K3,n with edges a0, b0, c0, a1, . . . , a(n − 1). T Let D be a drawing of K3,n in the plane such that |D(e) ∩ D(f)|2 = 0 if e ∈ T and e ∩ f = ∅. Claim 5. Either for all i = j, |D(bi) ∩ D(cj)| =2 0 iff (ai, aj) ∈ TD(a, a0); or for all i = j, |D(bi) ∩ D(cj)| =2 1 iff (ai, aj) ∈ TD(a, a0).

K3,n = ({a, b, c} ∪ {0, . . . , n − 1}, {a0, . . . , a(n − 1), b0, c0}) K3,n a c 1 2 3 . . . b a c 1 2 3 . . . b Let T be the spanning tree in K3,n with edges a0, b0, c0, a1, . . . , a(n − 1). T Let D be a drawing of K3,n in the plane such that |D(e) ∩ D(f)|2 = 0 if e ∈ T and e ∩ f = ∅. Claim 5. Either for all i = j, |D(bi) ∩ D(cj)| =2 0 iff (ai, aj) ∈ TD(a, a0); or for all i = j, |D(bi) ∩ D(cj)| =2 1 iff (ai, aj) ∈ TD(a, a0). Corollary 5. The rank of the n − 1 by n − 1 matrix M = (mij)

• ver Z2, where mij =2 |D(bi) ∩ D(cj)|, is at least

n−2

2

• .

K3,n = ({a, b, c} ∪ {0, . . . , n − 1}, {a0, . . . , a(n − 1), b0, c0}) a c 1 2 3 . . . b a c 1 2 3 . . . b Let T be the spanning tree in K3,n with edges a0, b0, c0, a1, . . . , a(n − 1). T Let D be a drawing of K3,n in the plane such that |D(e) ∩ D(f)|2 = 0 if e ∈ T and e ∩ f = ∅. Claim 6. Either for all i = j, |D(bi) ∩ D(cj)| =2 0 iff (ai, aj) ∈ TD(a, a0); or for all i = j, |D(bi) ∩ D(cj)| =2 1 iff (ai, aj) ∈ TD(a, a0). Corollary 6. The rank of the n − 1 by n − 1 matrix M = (mij) over Z2, where mij =2 |D(bi) ∩ D(cj)|, is at least

n−2

2

.

K3,n

K3,n = ({a, b, c} ∪ {0, . . . , n − 1}, {a0, . . . , a(n − 1), b0, c0}) a c 1 2 3 . . . b a c 1 2 3 . . . b Let T be the spanning tree in K3,n with edges a0, b0, c0, a1, . . . , a(n − 1). T Let D be a drawing of K3,n in the plane such that |D(e) ∩ D(f)|2 = 0 if e ∈ T and e ∩ f = ∅. Claim 6. Either for all i = j, |D(bi) ∩ D(cj)| =2 0 iff (ai, aj) ∈ TD(a, a0); or for all i = j, |D(bi) ∩ D(cj)| =2 1 iff (ai, aj) ∈ TD(a, a0). Corollary 6. The rank of the n − 1 by n − 1 matrix M = (mij) over Z2, where mij =2 |D(bi) ∩ D(cj)|, is at least

n−2

2

.

K3,n

• Proof. M + M T = In−1 + Jn−1, and thus,

rank(M T) + rank(M) ≥ rank(In−1 + Jn−1) = n − 2.

K3,n = ({a, b, c} ∪ {0, . . . , n − 1}, {a0, . . . , a(n − 1), b0, c0}) a c 1 2 3 . . . b a c 1 2 3 . . . b Let T be the spanning tree in K3,n with edges a0, b0, c0, a1, . . . , a(n − 1). T Let D be a drawing of K3,n in the plane such that |D(e) ∩ D(f)|2 = 0 if e ∈ T and e ∩ f = ∅. Claim 6. Either for all i = j, |D(bi) ∩ D(cj)| =2 0 iff (ai, aj) ∈ TD(a, a0); or for all i = j, |D(bi) ∩ D(cj)| =2 1 iff (ai, aj) ∈ TD(a, a0). Corollary 6. The rank of the n − 1 by n − 1 matrix M = (mij) over Z2, where mij =2 |D(bi) ∩ D(cj)|, is at least

n−2

2

.

K3,n

Corollary 7. If K3,n admits a Z2-embedding on a surface S then K3,n embeds on S.

• Proof. M + M T = In−1 + Jn−1, and thus,

rank(M T) + rank(M) ≥ rank(In−1 + Jn−1) = n − 2.

Unsolved problems

Unsolved problems Can we decide in a polynomial time if a given graph Z2-embeds

• n a given surface?

Unsolved problems Can we decide in a polynomial time if a given graph Z2-embeds

• n a given surface?

Computing the orientable Z2-genus is NP-hard. (Follows by the result of Thomassen showing NP-hardness for computing the

• rientable genus of cubic graphs.)

Unsolved problems Can we decide in a polynomial time if a given graph Z2-embeds

• n a given surface?

Computing the orientable Z2-genus is NP-hard. (Follows by the result of Thomassen showing NP-hardness for computing the

• rientable genus of cubic graphs.)

Does eg(G) = eg0(G) where eg(G) and eg0(G) is the Euler genus and Euler Z2-genus, respectively?(Conjecture by Schaefer and ˇ Stefankoviˇ c)