1. Hypergraphs and their planar embeddings Marisa Debowsky University of Vermont April 25, 2003

2. Things I Want You To Get Out Of This Lecture n The definition of a hypergraph. n Some understanding of the main question: “When is a hypergraph planar?” n The concept of a partial ordering on graphs. n Some understanding of the answer to the main question!

3. Definitions n A hypergraph is a generalization of a graph. An edge in a graph is defined as an (unordered) pair of vertices. In a hypergraph, an edge (or hyperedge) is simply a subset of the vertices (of any size). n The rank of a hyperedge is the number of vertices incident with that edge. The rank of the hypergraph H is the size of the largest edge of H.

4. Example 1 23 456 V(H) = {1, 2, 3, 4, 5, 6} E(H) = {124, 136, 235, 456}

5. Planar Graphs n A graph G is planar if there exists a drawing of G in the plane with no edge crossings. n Kuratowski gave necessary and sufficient conditions for a graph to be planar: Thm: A graph G is planar if, and only if, it contains no subdivision of K 3,3 or K 5.

6. Planar Hypergraphs? n In order to ask questions about planar hypergraphs, we need to make sure that the concept is well-defined.

7. Drawing a Hypergraph the long-winded definition Def n : A hypergraph H has an embedding (or is planar) if there exists a graph M such that V(M) = V(H) and M can be drawn in the plane with the faces two-colored (say, grey and white) so that there exists a bijection between the grey faces of M and the hyperedges of H so that a vertex v is incident with a grey face of M iff it is incident with the corresponding hyperedge of H.

8. Example 1 23 4 5 6 V(M) = {1, 2, 3, 4, 5, 6} E(M) = {12, 24, 14, 13, 36, 16, 23, 25, 35, 45, 56, 46} F(M) = {124, 136, 235, 456, 123, 245, 356, 146} V(H) = {1, 2, 3, 4, 5, 6} E(H) = {124, 136, 235, 456} F(H) = {123, 245, 356, 146}

9. Main Question n Which hypergraphs are planar? Can we find an obstruction set to planar hypergraphs (akin to K 3,3 and K 5 for planar graphs)? (Okay, that was more than one question.)

10. The Incidence Graph Given a hypergraph, H, we can construct a bipartite graph G derived from H. Let V 1 V 2 be the vertices of G. The vertices in V 1 correspond to V(H) and the vertices in V 2 correspond to E(H). A vertex v V 1 is adjacent to a vertex w V 2 if the corresponding hypervertex v is incident with the corresponding hyperedge w. Because the bipartite graph describes the incidences of the vertices and edges of H, we call G the incidence graph of H.

11. Example 2 1 3 456 2 1 3 456 In the bipartite graph on the right, the circled vertices correspond to hyperedges.

12. A Handy Reduction Theorem and the Main Question, again n Thm: A hypergraph is planar if and only if its incidence graph is planar. n This allows us to rephrase our question: Which bipartite graphs are planar?

13. Graphs Inside Graphs When we say that K 3,3 and K 5 are the “smallest” non-planar graphs or the “obstructions” to planarity, we mean that every non-planar graph contains a copy of K 3,3 or K 5 as a subgraph - in other words, contains of subdivision of K 3,3 or K 5. Can we formulate a notion similar to “subgraph” or “subdivision” for bipartite graphs that extends naturally to hypergraphs?

14. Partial Orderings n We can rank graphs using a partially ordered set: the set of all graphs together with a relation “< ” which is reflexive, antisymmetric, and transitive. Note: This is different from a “totally ordered set”!

15. Graph Operations n Frequently, we will form a graph G 2 from a graph G 1 where G 2 < G 1 by a modification called a graph operation. Different combinations of operations create distinct partial orderings of graphs. You are already familiar with some: deleting an edge from G 1, for example, creates a subgraph of G 1. n We will consider four different partial orders: detachment, bisubdivision, deflation, and duality.

16. Hereditary Properties n A property P is called hereditary under the partial order “ < ” if, whenever G P and H < G, it follows that H P. n Planarity is a hereditary property under these four operations, so we can consider the obstruction set to planarity under each operation.

17. Size of the Obstruction Sets n The detachment operation on hypergraphs corresponds to the subgraph operation in graphs: its obstruction set is infinite. n Adding the bisubdivision operation reduces the obstructions to a finite set, and each additional operation makes the set smaller.

18. Detachment Ordering n H is a detachment of G if it is obtained by removing an edge from the incidence graph. This corresponds to removing an incidence between a vertex and a hyperedge: pictorally, “detaching” a vertex from the hyperedge. Under the detachment ordering, H < G iff H is a detachment of G.

19. Detachment Example

20. Bisubdivision Ordering n H is a bisubdivision of G if it is formed by removing two interior degree-2 vertices from an edge of the incidence graph. This corresponds to contracting a hyperedge of rank 2. Under the bisubdivision ordering, H < G iff H is a bisubdivision or detachment of G.

21. Bisubdivision Example

22. Deflation Ordering n Suppose a bipartite graph G has a vertex of degree n from one partite set surrounded by (that is, adjacent to) n vertices of degree 2 from the other partite set. H is a deflation of G if it is obtained by removing those n vertices and reassigning the interior vertex (still of degree n) to the other partite set. In the hypergraph, this corresponds to “deflating” a hyperedge of rank n to a single vertex. Under the deflation ordering, H < G iff H is a deflation, bisubdivision, or detachment of G.

23. Deflation Example

24. Duality n The incidence graph is a bipartite graph; one partite set corresponds to the vertices of the hypergraph and the other to the hyperedges. Reversing the assignments of the partite sets produces a (generally) different hypergraph. n Def n : A hypergraph H is the dual of a hypergraph G if they are obtained from the same incidence graph.

25. Duality Ordering and Example n The duality ordering has H < G iff H is the dual of G. Bipartite Incidence Graph Hypergraph GHypergraph H

26. The Main Question... Again. n One more time: What are the obstructions to embedding bipartite graphs in the plane under each partial ordering?

27. The Answer! (for bipartite graphs) n Thm: There are exactly 9 non- planar bipartite graphs under the partial ordering of bisubdivision and detachment. The bipartite obstructions, G 1 - G 9, are given below.

28. Bipartite graphs G 1 - G 9 G1G1 G2G2 G3G3 G4G4 G5G5 G6G6 G7G7 G8G8 G9G9

29. The Answer! (for hypergraphs) n Corollary: There are exacly 16 non-planar hypergraphs under the partial ordering of bisubdivision and detachment. The hypergraph obstructions, H 1 - H 16, are given below.

30. Hypergraphs H 1 - H 16 H1H1 H2H2 H3H3 H4H4 H5H5 H6H6 H7H7 H8H8 H9H9 H10H10 H 11 H12H12 H13H13 H 14 H 15 H 16

31. Other Partial Orderings n Thm: There are exactly 2 non-planar bipartite graphs under the partial ordering of deflation, bisubdivision, and detachment. They are G 1 and G 4. n Corollary: There are exactly 3 non-planar hypergraphs under the partial ordering of deflation, bisubdivision, and detachment. They are H 1, H 2, and H 7.

32. Still More Partial Orderings n Thm: There are exactly 9 non-planar hypergraphs under the partial ordering of duality, bisubdivision, and detachment. They are H 1, H 3, H 5, H 7, H 8, H 9, H 11, H 13 and H 15. n Thm: There are exactly 2 non-planar hypergraphs under the partial ordering of duality, deflation, bisubdivision, and detachment. They are H 1 and H 7.

33. Further Research n Analogues of Kuratowski’s Theorem have been developed for other surfaces. Can we find the obstruction sets for embedding hypergraphs in, for example, the projective plane? n There are 2 non-planar graphs and 16 non- planar hypergraphs. There are 103 non- projective-planar graph, which leads us to suspect on the order of 800 non-projective- planar hypergraphs.

34. Contact Information n You can reach me at marisa.debowsky@uvm.edu or find me online at http://www.emba.uvm.edu/~mdebowsk/. n The work presented was done jointly with Professor Dan Archdeacon at UVM. You can reach him at dan.archdeacon@uvm.edu.

35. H1H1 H2H2 H3H3 H4H4 H5H5 H6H6 H7H7 H8H8 H9H9 H 10 H 11 H 12 H 13 H 14 H 15 H 16