Representing Graphs and Hypergraphs by Touching Polygons in 3D · Representing Graphs and Hypergraphs by Touching Polygons in 3D PawełRzążewski,NoushinSaeedi ... Complete 3-uniform

Representing Graphs and Hypergraphs byTouching Polygons in 3D

Paweł Rzążewski, Noushin Saeedi

joint work with William Evans, Chan-Su Shin, and Alexander Wolff

How to draw a graph? (in 2d)

I non-crossing drawings

→ planar graphs, polynomial-time

I contact representations

I intersection representations

I segments

→ SEG, ∃R-complete

I convex sets

→ CONV, ∃R-complete


I non-crossing drawings → planar graphs, polynomial-time



I segments


I convex sets






I segments


I convex sets






I segments → SEG, ∃R-completeI convex sets → CONV, ∃R-complete






Contact representations by polygons in 2d

I polygons are interior-disjointI at most two polygons touch in one point

I G admits a contact representation → G planar

I G planar → G admits a contact representation















I G admits a contact representation → G planarI G planar → G admits a contact representation















I contact representations → planar graphs, polynomial-time




I non-crossing drawings

→ every graph, trivial



I segments

→ ∃R-complete

I convex sets

Theorem. Recognizing segment intersection graphs in 3d is∃R-complete.


I non-crossing drawings → every graph, trivial



I segments

→ ∃R-complete

I convex sets






I segments → ∃R-completeI convex sets






I segments → ∃R-completeI convex sets


Contact representations by touching polygons

Theorem. Every graph can be represented by touching convexpolygons in 3d.

I in particular, this is an intersection representation by convexsets

Key lemma

Lemma. For every n ≥ 3 there is an arrangement of lines`1, `2, . . . , `n, such that:a) `i intersects `1, `2, . . . , `n in this ordering (pi ,j := `i ∩ `j),b) distances decrease exponentially: for every i , j we have

dist(pi ,j−1, pi ,j) ≥ 2dist(pi ,j , pi ,j+1).

Key lemma



`2

`1

`3

p12 p13

p23

Key lemma



`2

`1

`3

p12 p13

p23

p34

Key lemma



`2

`1

`3

`4p12 p13p14

p23

p34

Representing graphsI assume G is completeI set height of pi ,j to min(i , j)I vi is represented by convex hull of pi ,j ’s

I consider i < j : pi ,j is the touching pointI Pi and Pj are interior-disjoint

I for arbitrary graphs: if vivj is a non-edge, remove pi ,j from Piand Pj




pi,1

pi,i−2

pi,i−1

pi,i+1 pi,n

1

i− 1

i

i− 2


I consider i < j : pi ,j is the touching point

I Pi and Pj are interior-disjoint





Pi Pj

pi,j




Pi Pj

pi,j



I contact representations → every graph, non-trivial


I segments → ∃R-completeI convex sets → every graph, non-trivial

Grid size

I our representation requires exponential-sized gridI we consider also special classes of graphs

Graph class general bipartite 1-plane subcubiccubic

Grid volume super-poly O(n4) O(n2) O(n3)Running time O(n2) linear linear O(n log2 n)

Drawing Hypergraphs

Graph G = (V , E )

Polygons Contact points

Hypergraph H = (V , E )

Contact points Polygons

Drawing Hypergraphs

Graph G = (V , E )




Drawing Hypergraphs

Graph G = (V , E )




Complete 3-uniform HypergraphsA hypergraph is 3-uniform if all its hyperedges are of cardinality 3.

Theorem (Carmesin [ArXiv’19])Complete 3-uniform hypergraphs with n ≥ 6 vertices cannot berealized by non-crossing triangles in 3d.

I The link graph of a simplicial 2-complex at a vertex vhasI a node for every segment at v , andI an arc between two nodes if they share a face at v .

I If there is a non-crossing drawing, the link graph at anyvertex must be planar.

v





v





v





v





v





v

Steiner Systems

A Steiner system S(t, k, n) is an n-element set S togetherwith a set of k-element subsets of S, called blocks, such thateach t-element subset of S is contained in exactly one block.

Steiner Triple Systems1

S(2, 3, 7)

1 2 31 4 71 5 62 4 62 5 73 4 53 6 7

S(2, 3, 9)

1 2 3 1 5 94 5 6 2 6 77 8 9 3 4 81 4 7 1 6 82 5 8 2 4 93 6 9 3 5 7

Steiner Quadruple System

S(3, 4, 8)

1 2 4 8 3 5 6 72 3 5 8 1 4 6 73 4 6 8 1 2 5 74 5 7 8 1 2 3 61 5 6 8 2 3 4 72 6 7 8 1 3 4 51 3 7 8 2 4 5 6

1Ossona de Mendez [JGAA’02] shows that any 3-uniform hypergraph withincidence poset dimension 4 has a non-crossing drawing with triangles. This impliesthe existence of 3d representations (with exponential coordinates) for the two smallestSteiner triple systems.

Steiner Triple Systems

TheoremThe Fano plane S(2, 3, 7) has a non-crossingdrawing.

S(2, 3, 7)

1 2 31 4 71 5 62 4 62 5 73 4 53 6 7

5 72

3

14 6

2d drawing

1

2

64

7

3

5

top 3d view

1

26 4

73

5

side 3d view

Steiner Triple Systems (cont.)

side view

12

6

4

3

5

8

9

top view

1

2

6

4

3

5

8

9

S(2, 3, 9)

1 2 34 5 67 8 91 4 72 5 83 6 91 5 92 6 73 4 81 6 82 4 93 5 7

TheoremThe Steiner triple system S(2, 3, 9) has a non-crossing drawing.


side view

12

6

4

3

5

8

9

top view

1

2

6

4

3

5

8

9

S(2, 3, 9)

1 2 34 5 67 8 91 4 72 5 83 6 91 5 92 6 73 4 81 6 82 4 93 5 7



side view

12

6

4

3

5

8

9

top view

1

2

6

4

3

5

8

9

S(2, 3, 9)

1 2 34 5 67 8 91 4 72 5 83 6 91 5 92 6 73 4 81 6 82 4 93 5 7



side view

12

6

4

3

5

8

9

top view

1

2

6

4

3

5

8

9

S(2, 3, 9)

1 2 34 5 67 8 91 4 72 5 83 6 91 5 92 6 73 4 81 6 82 4 93 5 7



side view

12

6

4

3

5

8

9

top view

1

2

6

4

3

5

8

9

S(2, 3, 9)

1 2 34 5 67 8 91 4 72 5 83 6 91 5 92 6 73 4 81 6 82 4 93 5 7



side view

12

6

4

3

5

8

9

top view

1

2

6

4

3

5

8

9

S(2, 3, 9)

1 2 34 5 67 8 91 4 72 5 83 6 91 5 92 6 73 4 81 6 82 4 93 5 7

14

26

7

358

9

Steiner Quadruple SystemsTheoremThe Steiner quadruple system S(3, 4, 8) does not have anon-crossing drawing.

1248 36 P1236 ∩ P3468 = l36 and l12 ∩ l48 ∈ l36

1248 37 P1378 ∩ P2347 = l37 and l18 ∩ l24 ∈ l37

1248 67 P1467 ∩ P2678 = l67 and l14 ∩ l28 ∈ l67

S(3, 4, 8)

1 2 4 8 3 5 6 72 3 5 8 1 4 6 73 4 6 8 1 2 5 74 5 7 8 1 2 3 61 5 6 8 2 3 4 72 6 7 8 1 3 4 51 3 7 8 2 4 5 6

If there is a drawing,I 3,6, and 7 are all placed at the same point.I 3567 is degenerate; a contradiction.

(In fact, we can show that 3567 is just a point.)

TheoremThe Steiner quadruple system S(3, 4, 10) cannot be drawn using allconvex or all non-convex non-crossing quadrilaterals.


1248 36 P1236 ∩ P3468 = l36 and l12 ∩ l48 ∈ l36

1248 37 P1378 ∩ P2347 = l37 and l18 ∩ l24 ∈ l37

1248 67 P1467 ∩ P2678 = l67 and l14 ∩ l28 ∈ l67

S(3, 4, 8)

1 2 4 8 3 5 6 72 3 5 8 1 4 6 73 4 6 8 1 2 5 74 5 7 8 1 2 3 61 5 6 8 2 3 4 72 6 7 8 1 3 4 51 3 7 8 2 4 5 6





1248 36 P1236 ∩ P3468 = l36 and l12 ∩ l48 ∈ l36

1 4

2

8

3

6

S(3, 4, 8)

1 2 4 8 3 5 6 72 3 5 8 1 4 6 73 4 6 8 1 2 5 74 5 7 8 1 2 3 61 5 6 8 2 3 4 72 6 7 8 1 3 4 51 3 7 8 2 4 5 6





1248 36 P1236 ∩ P3468 = l36 and l12 ∩ l48 ∈ l36

1248 37 P1378 ∩ P2347 = l37 and l18 ∩ l24 ∈ l37

1248 67 P1467 ∩ P2678 = l67 and l14 ∩ l28 ∈ l67

S(3, 4, 8)

1 2 4 8 3 5 6 72 3 5 8 1 4 6 73 4 6 8 1 2 5 74 5 7 8 1 2 3 61 5 6 8 2 3 4 72 6 7 8 1 3 4 51 3 7 8 2 4 5 6





1248 36 P1236 ∩ P3468 = l36 and l12 ∩ l48 ∈ l36

1248 37 P1378 ∩ P2347 = l37 and l18 ∩ l24 ∈ l37

1248 67 P1467 ∩ P2678 = l67 and l14 ∩ l28 ∈ l67

S(3, 4, 8)

1 2 4 8 3 5 6 72 3 5 8 1 4 6 73 4 6 8 1 2 5 74 5 7 8 1 2 3 61 5 6 8 2 3 4 72 6 7 8 1 3 4 51 3 7 8 2 4 5 6




Steiner Quadruple Systems (cont.)TheoremNo Steiner quadruple system can be drawn using convexquadrilaterals2.I Any vertex v is incident to (n−1)(n−2)

6 quadrilaterals.I Add the diagonals incident to v to get a simplicial 2-complex.I The link graph at v has (n−1)(n−2)

3 edges and n − 1 vertices.I For n > 8, the link graph is not planar.

TheoremNo Steiner quadruple system with 20 or morevertices can be drawn using quadrilaterals.

ConjectureNo Steiner quadruple system can be drawn using non-crossingquadrilaterals.

2We thank Arnaud de Mesmay and Eric Sedgwick for pointing us to alemma of Dey and Edelsbrunner [DCG’94], which uses the same proof idea.













Open problems

Other hypergraphs Larger Steiner triple systems/projectiveplanes.

Hardness Is deciding whether a 3-uniform hypergraphhas a non-crossing drawing with trianglesNP-hard?

Grid size Can any graph be represented with convexpolygons on a polynomial sized grid?

Nicer drawings Small aspect ratio, large angle resolution, etc.

Documents

Representing Graphs and Hypergraphs by Touching Polygons in 3D · Representing Graphs and Hypergraphs by Touching Polygons in 3D PawełRzążewski,NoushinSaeedi ... Complete 3-uniform