Hypergraphsand their planar embeddings

Marisa DebowskyUniversity of Vermont

April 25, 2003

Things I Want You To Get Out Of This Lecture 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 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.

Example1

2 3

4 5 6

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

Planar Graphs 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 K3,3 or K5.

Planar Hypergraphs?

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

Drawing a Hypergraphthe 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

1

2 3

4 56

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}

Main Question

Which hypergraphs are planar? Can we find an obstruction set to planar hypergraphs (akin to K3,3 and K5 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 V1 V2 be the vertices of G. The vertices in V1 correspond to V(H) and the vertices in V2 correspond to E(H). A vertex v V1 is adjacent to a vertex w V2 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.

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Example

2

1

3

4 5 6

2

1

3

4 5 6

In the bipartite graph on the right, the circled vertices correspond to hyperedges.

A Handy Reduction Theoremand the Main Question, again

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 K3,3 and K5 are the “smallest” non-planar graphs or the “obstructions” to planarity, we mean that every non-planar graph contains a copy of K3,3 or K5 as a subgraph - in other words, contains of subdivision of K3,3 or K5.

Can we formulate a notion similar to “subgraph” or “subdivision” for bipartite graphs that extends naturally to hypergraphs?

Partial Orderings

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

Frequently, we will form a graph G2 from a graph G1 where G2 < G1 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 G1, for example, creates a subgraph of G1.

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

Hereditary Properties

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.

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Size of the Obstruction Sets

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 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 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 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

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

The duality ordering has H < G iff H is the dual of G.

Bipartite Incidence Graph

Hypergraph G Hypergraph H

The Main Question... Again.

One more time:

What are the obstructions to embedding bipartite graphs in the plane under each partial ordering?

The Answer! (for bipartite graphs)

Thm: There are exactly 9 non-planar bipartite graphs under the partial ordering of bisubdivision and detachment.

The bipartite obstructions, G1 - G9, are given below.

Bipartite graphs G1 - G9

G1 G2 G3

G4 G5 G6

G7 G8 G9

The Answer! (for hypergraphs)

Corollary: There are exacly 16 non-planar hypergraphs under the partial ordering of bisubdivision and detachment.

The hypergraph obstructions, H1 - H16, are given below.

Hyp

ergr

aphs

H1 -

H16

H1 H2 H3 H4

H5 H6 H7 H8

H9 H10 H11 H12

H13 H14H15 H16

Other Partial Orderings

Thm: There are exactly 2 non-planar bipartite graphs under the partial ordering of deflation, bisubdivision, and detachment. They are G1 and G4.

Corollary: There are exactly 3 non-planar hypergraphs under the partial ordering of deflation, bisubdivision, and detachment. They are H1, H2, and H7.

Still More Partial Orderings Thm: There are exactly 9 non-planar

hypergraphs under the partial ordering of duality, bisubdivision, and detachment. They are H1, H3, H5, H7, H8, H9, H11, H13 and H15.

Thm: There are exactly 2 non-planar hypergraphs under the partial ordering of duality, deflation, bisubdivision, and detachment. They are H1 and H7.

Further Research

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-projective-planar graph, which leads us to suspect on the order of 800 non-projective-planar hypergraphs.

Contact Information

You can reach me at marisa.debowsky@uvm.edu

or find me online athttp://www.emba.uvm.edu/~mdebowsk/.

The work presented was done jointly with Professor Dan Archdeacon at UVM. You can reach him at dan.archdeacon@uvm.edu.

H1 H2 H3 H4

H5 H6 H7 H8

H9 H10 H11 H12

H13 H14H15 H16