representing graphs and hypergraphs by touching polygons in 3d · representing graphs and...
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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
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
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
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-completeI convex sets → CONV, ∃R-complete
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-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
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
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
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
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 planarI G planar → G admits a contact representation
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 planarI G planar → G admits a contact representation
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 planarI G planar → G admits a contact representation
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 planarI G planar → G admits a contact representation
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 planarI G planar → G admits a contact representation
How to draw a graph? (in 2d)
I non-crossing drawings → planar graphs, polynomial-time
I contact representations → planar graphs, polynomial-time
I intersection representations
I segments → SEG, ∃R-completeI convex sets → CONV, ∃R-complete
How to draw a graph? (in 3d)
I non-crossing drawings
→ every graph, trivial
I contact representations
I intersection representations
I segments
→ ∃R-complete
I convex sets
Theorem. Recognizing segment intersection graphs in 3d is∃R-complete.
How to draw a graph? (in 3d)
I non-crossing drawings → every graph, trivial
I contact representations
I intersection representations
I segments
→ ∃R-complete
I convex sets
Theorem. Recognizing segment intersection graphs in 3d is∃R-complete.
How to draw a graph? (in 3d)
I non-crossing drawings → every graph, trivial
I contact representations
I intersection representations
I segments → ∃R-completeI convex sets
Theorem. Recognizing segment intersection graphs in 3d is∃R-complete.
How to draw a graph? (in 3d)
I non-crossing drawings → every graph, trivial
I contact representations
I intersection representations
I segments → ∃R-completeI convex sets
Theorem. Recognizing segment intersection graphs in 3d is∃R-complete.
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
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).
`2
`1
`3
p12 p13
p23
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).
`2
`1
`3
p12 p13
p23
p34
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).
`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
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
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 point
I Pi and Pj are interior-disjoint
I for arbitrary graphs: if vivj is a non-edge, remove pi ,j from Piand Pj
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 Pj
pi,j
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 Pj
pi,j
How to draw a graph? (in 3d)
I non-crossing drawings → every graph, trivial
I contact representations → every graph, non-trivial
I intersection representations
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 )
Polygons Contact points
Hypergraph H = (V , E )
Contact points Polygons
Drawing Hypergraphs
Graph G = (V , E )
Polygons Contact points
Hypergraph H = (V , E )
Contact points Polygons
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
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
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
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
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
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
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.
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.
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.
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.
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.
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
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.
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.
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
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
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.
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.
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.
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.
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.
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.