Hypergraphs and their planar embeddings Marisa Debowsky University

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

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

Definitions n 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). 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.

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

Planar Graphs n n A graph G is planar if there exists a drawing of G in the plane with no edge crossings. 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.

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

Drawing a Hypergraph the long-winded definition Defn: 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.

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

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. )

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.

Example 1 2 4 1 3 5 2 6 4 3 5 6 In the bipartite graph on the right, the circled vertices correspond to hyperedges.

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

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?

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”!

Graph Operations n 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. We will consider four different partial orders: detachment, bisubdivision, deflation, and duality.

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

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

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.

Detachment Example

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.

Bisubdivision Example

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.

Deflation Example

Duality n 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. Defn: A hypergraph H is the dual of a hypergraph G if they are obtained from the same incidence graph.

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

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

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

Bipartite graphs G 1 G 9 G 1 G 2 G 3 G 4 G 5 G 6 G 7 G 8 G 9

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.

Hypergraph s H 1 - H 16 H 1 H 5 H 9 H 13 H 2 H 3 H 6 H 10 H 14 H 7 H 8 H 11 H 12 H 15 H 16

Other Partial Orderings n 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. 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.

Still More Partial Orderings n 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. 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.

Further Research n 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? There are 2 non-planar graphs and 16 non-planar hypergraphs. There are 103 non-projectiveplanar graph, which leads us to suspect on the order of 800 non-projective-planar hypergraphs.

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.

H 1 H 5 H 9 H 13 H 2 H 3 H 6 H 10 H 14 H 7 H 8 H 11 H 12 H 15 H 16