1. More on graph drawings other representations Alex Sludnikov Seminar in Computational Geometry

2. Some conjectures Scheinerman ’ s Problem Planar graph G Can we represent it by line segments in the plane so that 2 segments intersect  the corresponding vertices are adjacent in G s.

3. But what do we know? We know that it ’ s true for bipartite graphs. Sufficient to use vertical and horizontal segments (2 directions). This result was extended to 3-colorable planar graphs (but 3 directions are not always sufficient). In fact, we don ’ t know if a representation by contiguous arcs always exists, with the property that 2 arcs cannot intersect more than once.

4. Harborth ’ s conjecture Every planar graph admits a straight line drawing in which the length of every edge is an integer. 4 3 4 3 5

5. Papadimitriou-Ratajczak G – 3-connected planar graph. G can be drawn in the plane with possibly crossing straight-line edges so that for any u,v V there is a path from u to v along which the Euclidean distance to v strictly decreases. It ’ s not hard to show that K 2,11 and K 3,16 cannot be drawn with the required property.

6. Why it doesn ’ t work for K 2,11? (intuition) Lets mark the 2 vertices (of the first side) u,v and the rest as a1, a2, …, a11 Suppose we want to draw this graph so it works for the path that start at u and goes through all the a ’ s. The points are getting crowder so eventually we won ’ t be able to draw straight lines.

7. The rectangular dual Let ’ s extend our representation of horizontal and vertical segments to rectangles. Two rectangles are tangent  the corresponding vertices are adjacent in G. No 4 rectangles share a vertex.

8. Theorem A planar graph admits a rectangular dual if and only if: Its exterior face is a quadrilateral. Its interior faces are triangles. It has no seperating triangle.

9. An example N W S E a b c f g d e h

10. The dual a b c d e f g h

11. A graph G satisfying the conditions can be directed and split into 2 edge-disjoint subgraphs G h, G v (h – horizontal, v-vertical). N W E a b c f g d e h S

12. Denote the faces in Gh (Gv) seperating the in and out edges at an internal vertex u by lower(u) and upper(u) (left(u) and right(u) in Gv). let d(f) denote the length of the largest chain of faces connecting the leftmost (uppermost) face to f. In the dual rectangle representation we set the coordinates the following way: ( d(left(u), d(lower(u) )( d(right(u), d(lower(u) ) ( d(left(u), d(upper(u) )( d(right(u), d(upper(u) ) u Constructing the Dual

13. So let ’ s try to number the rectangles. Let ’ s give label 1 to the rectangle containing the lower left corner of the big rectangle. Notice the following: One of the following options holds: 1. The right side of the rectangle labeled 1 is full segment in the tiling. 2. The upper side of the rectangle labeled 1 is full segment in the tiling.

14. Extracting a permutation In the first case let ’ s shrink rectangle 1 into a vertical segment. In the second case we will shrink it into the horizontal segment. Now we get a new rectangle (lets label it 2) in the lower corner. Let ’ s continue recursively … We can do it on the upper left rectangle as well. So let ’ s write the sequence of the labels as they disappear.

15. A Demonstration 1 4 3 2 5

16. Step 1 1 3 2 5 The permutation: 4

17. Step 2 3 2 5 The permutation: 4, 1

18. Step 3 2 5 The permutation: 4, 1, 3

19. Step 4 2 The permutation: 4, 1, 3, 5

20. In the end we got 4, 1, 3, 5, 2 Why was it any good? Lemma Let i<j be two labels. If i precedes j in the permutation then rectangle i is completely to the left of rectangle j. If j precedes i in the permutation then rectangle i is completely below rectangle j. Conclusion Such permutation uniquely determines the combinatorial type of the tiling.

21. The Crossing Number Definition For graph G, the crossing number cr(G) is the smallest number of edge crossings in any drawing of G. Clearly, cr(G)=0  G is planar. Finding the crossing number is an NP-complete problem. Of course we can look at specific graphs and check their cr(G)

22. Examples The utility graph (cr(G)=1) The Petersen graph (cr(G)=2) The Heawood Graph (cr(G)=3)

23. Some Special Cases Conjecture 1 The crossing number of the complete bipartite graph K n,m satisfies Conjecture 2 The crossing number of the complete graph K n satisfies The best upper bounds are the values in conjecture 1,2 The best lower bounds are

24. Example n=5 m=6 So we get

25. Again Conjectures Let C n denote a cycle with n vertices For any 2 graphs G and H, let G X H denote the graph on vertex set V(G) X V(H), where every vertex (x,y), x V(G), y V(H), is connected to all vertices (x ’,y ’ ) such that x=x ’ and yy ’ E(H), or xx ’ E(G) and y=y ’. For every n≥m≥3, we have It was proved for m≥3, n≥m(m+1)

26. More bounds and Limits The crossing lemma For any graph G with n vertices and e≥7n edges we have We know that 0.032 < 1024/31827 ≤ K0 ≤ 0.09. Problem: determine the precise value of K0 Do there exist positive constants C 1 and C 2 such that for any function e = e(n) satisfying C 1 n ≤ e ≤ C 2 n^2, we have

27. The Bisection Width Intuition Let G be a graph with vertex set V (G) and edge set E(G). The bisection width b(G) of G is defined as the minimum number of edges whose removal splits the graph into two roughly equal subgraphs. More precisely, b(G) is the minimum number of edges between V 1 and V 2 over all partitions of the vertex set of G into two disjoint parts V1 ∪ V2 such that |V 1 |, |V 2 | ≥ |V(G)|/3 We know that (d i are the degrees):

28. Just to show what I mean Lets look at the following graph And indeed

29. Monotone Property A graph property P is said to be monotone if (a) for any graph G satisfying P, every subgraph of G also satisfies P, and (b) if G1 and G2 satisfy P, then their disjoint union also satisfies P. For any monotone property P, let ex(n,P) denote the maximum number of edges that a graph of n vertices can have if it satisfies P. Let P be a monotone graph property with there exist two constants c, c′ > 0 such that the crossing number of any graph G with property P that has n vertices and edges satisfies

30. Other Crossing Numbers The rectilinear crossing number lin-cr(G) of a graph G is the minimum number of crossings in a drawing of G in which every edge is represented by a line segment. linr-cr(G)=0

31. The pairwise crossing number of G, pair-cr(G) is the minimum number of crossing pairs of edges over all drawings of G. (Here the edges can be represented by arbitrary continuous curves, so that two edges may cross more than once, but every pair of edges contributes at most one to pair-cr(G).) pair-cr(G)=1

32. The odd-crossing number of G, odd-cr(G) is the minimum number of those pairs of edges that cross an odd number of times over all drawings of G. odd-cr(G)=1

33. Some Properties And Problems odd-cr(G) ≤ pair-cr(G) ≤ cr(G) ≤ lin-cr(G) Hanani and Tutte ’ s theorem: a graph G is planar if and only if it can be drawn in the plane so that any two arcs representing edges of G cross an even number of times. Therefore, if odd-cr(G)=0 then it can be drawn with even number of crossings. Therefore, cr(G)=0. Problem: Does odd-cr(G) = pair-cr(G) = cr(G) hold for every graph G?

34. More Problems Given a graph G of n vertices and an integer K, can one check in polynomial time whether lin-cr(G) ≤ K? Does there exist a constant C such that cr(G) ≤ C (odd-cr(G)) holds for every graph G? The things we know:

35. Some Rules Each of the above crossing numbers with the exception of lin-cr(G) can further modified by applying one of the following rules: Rule + : Consider only those drawings of G in which two edges with a common endpoint do not cross each other. Rule 0 : Two edges with a common endpoint are allowed to cross, and their crossing counts. Rule − : Two edges with a common endpoint are allowed to cross, but their crossing does not count. In the original definitions we have always used Rule 0.

36. So what do we get? This gives us an array of nine different crossing numbers. It is easy to see that in a drawing of a graph that minimizes the number of crossing points, any two edges have at most one point in common Therefore, cr+(G) = cr(G), which slightly simplifies the picture.

37. From that we can obtain for any graph G with n vertices and with e ≥ 4n edges. Problem Do there exist suitable functions f1, f2, f3 such that every graph G satisfies: odd-cr(G) ≤ f1(odd-cr_(G)) pair-cr(G) ≤ f2(pair-cr_(G)) cr(G) ≤ f3(cr_(G)) ?

38. Thickness of a graph The thickness of a graph G is the minimum number of edge-disjoint planar graphs into which G can be partitioned. Geometric thickness of G is the same, except that now we consider only straight-line drawings. The thickness of a graph can never exceed its geometric thickness. Let ’ s look at graphs with 2 thickness.

39. An Example (3 Thickness)

40. Biplanar Crossing Number the minimum of cr(G1)+cr(G2) over all partitions of the graph into two edge-disjoint subgraphs, G1 and G2. The biplanar crossing number of a graph is 0 if and only if its thickness is at most 2. biplanar crossing number ≤ crossing number cause in the biplanar case we don ’ t care about the crossings between the edges of G1 and G2.

41. Convex Crossing Number Convex drawing Place the vertices of a graph G on a circle and connect the corresponding points by line segments. The convex crossing number of G is the minimum number of crossings in a convex drawing of G. A B C D E A B C D E

42. The convex crossing Lets look at the following example (K 4 ) And of course in this case it ’ s 1 Can be bounded above by a constant times of

43. Biplanar Convex Crossing Number Suppose we have a convex drawing of G so let ’ s check it ’ s biplanar number. The minimum on all the drawings is the Biplanar Convex crossing number. Another perspective: let ’ s color it ’ s edges in two colors, and look at the number of crossings of edges of the same color. The biplanar convex crossing number is the minimum number of such crossings in any convex drawing of G. biplanar convex crossing number ≥ biplanar crossing number biplanar convex crossing number ≤ convex crossing Is it true that the biplanar convex crossing number of every graph G with n vertices having degrees d 1,...,d n is at most ?

44. Let ’ s define some graphs Topological graph A graph G who ’ s edges are non-self-intersecting continuous arcs and not passing through any other point representing a vertex. u v

45. If the vertex set V (G) of a topological graph G is in general position, i.e., no three points that represent vertices are collinear and the edges are drawn by straight-line segments, then G is called a geometric graph.

46. Let G be a geometric graph, V (G) is the vertex set of a convex polygon, then G is called a convex geometric graph

47. Thrackles Topological graph with any pair of nonadjacent edges crossing precisely once Conway ’ s thrackle conjecture: The number of edges of a thrackle cannot exceed the number of its vertices.

48. More on Conway ’ s conjecture A thrackle of n vertices has fewer than 2n edges Actually, the bound is tighter: 3(n−1)/2 Conway ’ s conjecture is known to be true for straight- line thrackles A bipartite graph can be drawn as a thrackle if and only if it is planar.

49. For straight line trackles Theorem If any two edges of a geometric graph intersect (in an endpoint or an internal point), then it can have at most as many edges as vertices. Proof We say that an edge uv of a geometric graph is a leftmost edge at its endpoint u if turning the ray uv around u through 180 degrees in the counterclockwise direction, it never contains any other edge uw. In this case u is the leftmost u v w z

50. The proof For each vertex u, delete the leftmost edge at u, if such an edge exists. We claim that at the end of the procedure, no edge is left. if at least one edge uv remains, it follows that we did not delete it when we visited u and we did not delete it when we visited v. So there are vertices w, z such that Therefore wu and vz don ’ t intersect – contradiction! w u z v

51. Forbidden Geometric Subgraphs Given a class H of so-called forbidden geometric subgraphs, what is the maximum number of edges that a geometric graph G of n vertices can have without containing a geometric subgraph belonging to H? For instance, what is the maximum number of edges of a geometric graph with n vertices containing no k disjoint edges?

52. Problems in this field Problem Suppose a geometric graph G of n vertices containing no 4 disjoint edges. What is the smallest constant c such that the maximum number of edges is cn+O(n)? Problem Does there exist an absolute constant c such that every geometric graph with n vertices and no k disjoint edges has at most ckn edges? For convex geometric graphs the question is yes

53. Problems in this field Why? It follows from an old result of Perles (that if a convex geometric graph of n ≥ 2k vertices has more then (k − 1)n edges, it contains a noncrossing path of length 2k−1). This bound cannot be improved. Obviously, every noncrossing path of length 2k −1 has k disjoint edges. We call a topological graph simple if any two of its edges meet at most once Every thrackle is a simple topological graph.

54. Another Problem Problem Let ’ s take k≥3 Let ’ s look at a simple topological graph with n vertices that contains no k disjoint edges. Is it true that the maximum number of edges is O(n)?

55. Pairwise crossing edges A graph G has k pairwise edges if we can take k edges, so that each one of them intersects with the other. Problem Does there exist for every k > 3 a constant c k such that any geometric graph of n vertices containing no k pairwise crossing edges has at most c k n edges? For convex geometric graphs G with n ≥ 2k vertices and no k pairwise crossing edges we know that

56. For Example

57. Ramsey-Type Problems In classical Ramsey theory, one wants to find large monochromatic subgraphs in a complete graph whose edges are colored with several colors Most questions of this type can be generalized to complete geometric graphs, where the monochromatic subgraphs are required to satisfy certain geometric conditions.

58. Problem Does there exist for every k ≥ 3 a constant d k with the property that the edges of every geometric (or topological) graph with no k pairwise crossing edges can be colored by d k colors so that no two edges of the same color cross each other? We know it ’ s true for simple graph and k=3.

59. Another Problem Problem Does there exist a constant c > 0 such that every complete geometric graph with n vertices has at least cn pairwise crossing edges? one can always find at least c√n pairwise crossing edges

60. What we know if the edges of a complete geometric graph with n vertices are colored by two colors, then one can find a noncrossing spanning tree all of whose edges are of the same color ⌊ (n+1)/3 ⌋ pairwise disjoint edges of the same color. For example: 1 Black pairwise disjoint edges 1 Red pairwise disjoint edges

61. What we know For any two-coloring of the edges of a complete geometric graph with n vertices, there exists a noncrossing (simple) path of length Ω(2n/3) all of whose edges are of the same color.

62. What we don ’ t know Conjecture For every k ≥ 2 there exists a constant c k > 0 such that if the edges of a complete geometric graph with n vertices are colored by k colors, one can always find a monochromatic noncrossing path of length at least ckn. For k = 2, this conjecture has been verified for convex geometric graphs. Here one can find a monochromatic noncrossing path of length ⌊ (n+1)/2 ⌋.

63. Other Approaches Problem Determine the largest number c such that for any n red points and n blue points in general position in the plane, one can always find a noncrossing alternating path of length at least cn. Geometric graphs can also be regarded as families of segments. Problem What is the largest integer s(n) such that any family of n closed segments in general position in the plane has s(n) members that are neither pairwise disjoint or pairwise crossing?

64. It is known that

65. Geometric Hypergraphs We want to generalize the results on geometrical graphs to higher dimensions A geometric r-hypergraph is a pair (V,E), where V is a set of points in general position in d- space, and E is a set of closed (r−1)-dimensional simplices induced by some r-tuples of V. The sets V and E are called the vertex set and the (hyper)edge set of. a geometric graph is a two-dimensional geometric two-hypergraph

66. More Definitions - A class of geometric hypergraphs not permitted to be contained in the geometric hypergraphs under consideration is called a family of forbidden subhypergraphs. - the maximum number of edges that a d-dimensional geometric r-hypergraph of n vertices can have without containing a geometric subhypergraph belonging to

67. Theorem - the class of all geometric r-hypergraphs consisting of k pairwise disjoint edges Let V = V 1 ∪... ∪ V d (|V 1 |=... = |V d | = n) be a d n -element set in general position in d-space, and let E consist of all (d−1)-dimensional simplices having exactly one vertex in each Vi. Then E contains n disjoint simplices. This result can be applied to deduce that

68. Some conjectures - the class of all geometric r-hypergraphs consisting of k pairwise crossing edges.

69. Another one … We know that it ’ s true for k = 2 with ε = 0.