Subdivision Drawings of Hypergraphs Michael Kaufmann1, Marc van Kreveld2, and Bettina Speckmann3 1 Institut f¨ur Informatik, Universit¨at T¨ubingen [email protected] 2 Department of Computer Science, Utrecht University [email protected] 3 Department of Mathematics and Computer Science, TU Eindhoven [email protected] Abstract. We introduce the concept of subdivision drawings of hyper- graphs. In a subdivision drawing each vertex corresponds uniquely to a face of a planar subdivision and, for each hyperedge, the union of the faces corresponding to the vertices incident to that hyperedge is con- nected. Vertex-based Venn diagrams and concrete Euler diagrams are both subdivision drawings. In this paper we study two new types of sub- division drawings which are more general than concrete Euler diagrams and more restricted than vertex-based Venn diagrams. They allow us to draw more hypergraphs than the former while having better aesthetic properties than the latter. 1 Introduction AgraphG is a pair G =(V,E), where V is a set of elements or vertices and E ⊆ V × V is a set of pairs of vertices, called edges. A hypergraph H =(V,E)is a generalization of a graph, where again V is a set of elements or vertices, but E is a set of non-empty subsets of V , called hyperedges [1]. The set E ⊆P(V ) of hyperedges is a subset of the powerset of V . Hypergraphs are not as common as graphs, but they do arise in many appli- cation areas. In relational databases there is a natural correspondence between database schemata and hypergraphs, with attributes corresponding to vertices and relations to hyperedges [7]. Hypergraphs are used in VLSI design for circuit visualization [6, 14] and also appear in computational biology [10, 12] and social networks [3]. Drawings of hypergraphs are less well-understood than drawings of graphs. There is no single “standard” method of drawing hypergraphs, comparable to the point-and-arc drawings for graphs. When drawing hypergraphs, vertices are usually depicted as points or regions in the plane, but hyperedges can have very varied forms, including Steiner trees, closed curves in the plane, faces of subdivisions, and points. As a result, there is no unique definition of planarity This research was initiated during the Bertinoro Workshop on Graph Drawing, 2008. Bettina Speckmann is supported by the Netherlands Organisation for Scientific Re- search (NWO) under project no. 639.022.707. for hypergraphs—different drawing methods imply different, non-equivalent pla- narity definitions. In the following, we describe some of the drawing methods for hypergraphs in more detail. Hypergraph Drawings. The two most common methods to draw hypergraphs are the subset-based and the edge-based method. A subset-based drawing high- lights the fact that a hypergraph can be interpreted as a set system: Vertices are drawn as points in the plane and hyperedges are drawn as simple closed curves that contain exactly those vertices that they are incident to (see Fig. 1(a)). It is easy to see that any hypergraph can be drawn this way. Subset-based drawings neither know any concept of or experience any problems with planarity. Bertault and Eades [2] show how to create subset-based hypergraph drawings. An edge-based hypergraph drawing resembles standard drawings of graphs more closely [11]. Vertices are again drawn as points, but hyperedges are drawn as Steiner trees (see Fig. 1(b)). Edge-based drawings imply the most common definition for hypergraph planarity: A hypergraph is planar if and only if it has an edge-based drawing without hyperedge crossings. Another way to draw hypergraphs is the so-called Zykov representation. Ver- tices are again drawn as points, but hyperedges are visualized by faces of a subdivision. The vertices around a face of the subdivision are the ones con- nected by the corresponding hyperedge (see Fig. 1(c)). To distinguish faces that represent hyperedges from faces that do not, a background color is needed to fill all faces of the subdivision that do not correspond to hyperedges. A hypergraph is Zykov-planar if it has a Zykov representation. Zykov-planarity is equivalent to hypergraph planarity as induced by edge-based drawings. A hypergraph H can also be drawn as a bipartite graph where one set of vertices corresponds to the vertices of H and the other set corresponds to the hyperedges (see Fig. 1(d)). The edges of the bipartite graph represent vertex- hyperedge incidences. Also this representation immediately implies a definition of planarity, which is equivalent to Zykov-planarity and hence to hypergraph planarity as induced by edge-based drawings [17]. Many hypergraphs are not (Zykov-)planar and hence have neither an edge- based drawing, nor a Zykov representation, nor a drawing as a planar bipartite incidence graph. Motivated by this fact, Pollak and Johnson [9] study alternative 1 (1, 2) 12 2 2 1 1 2 (1, 2, 3) 3 4 (2, 4) 3 4 3 4 3 4 5 56565 6 (3, 4, 5, 6) (a) (b) (c) (d) 6 Fig. 1. Four drawings of the hypergraph H =(V,E)withV = {1, 2, 3, 4, 5, 6} and E = {(1, 2), (1, 2, 3), (2, 4), (3, 4, 5, 6)}: (a) subset-based drawing; (b) edge-based drawing; (c) Zykov representation; (c) incidence representation (bipartite graph). 4545 (1, 2) (1, 3) (1, 4) (1, 5) (1, 4) (1, 5) (1, 6) (2, 3) (2, 4) (2, 5) (2, 4) (2, 5) (2, 6) 12 12 (3, 4) (3, 5) (3, 4)(3, 5) (3, 6) 3 3 (a) (4, 5) (b) (c) 6 6 Fig. 2. A hyperedge-based Venn diagram for (a) K5 and (b) K3,3; (c) two different drawings of a vertex-based Venn diagram for H. definitions of hypergraph planarity which are implied by yet two more methods to draw hypergraphs—hyperedge-based Venn diagrams and vertex-based Venn di- agrams. (The choice of these names is somewhat unfortunate, since the drawings have little in common with standard Venn diagrams.) In a hyperedge-based Venn diagram, hyperedges correspond uniquely to the faces of a planar subdivision in such a way that for any vertex v, the union of the faces corresponding to hyperedges that contain v, is connected. In a vertex-based Venn diagram, vertices correspond uniquely to faces of a planar subdivision in such a way that for any hyperedge, the union of the faces corresponding to its incident vertices is connected. Note that the subdivision itself does not show the hyperedges directly, curves that enclose unions of faces must still be drawn to visualize them. A hypergraph is hyperedge-planar or vertex-planar if a planar subdivision with the required properties exists. It is easy to see that both K5 and K3,3 are hyperedge-planar (see Fig. 2(a, b)). Concerning vertex-planarity, consider the following hypergraph H: H has six vertices and three hyperedges (1, 2, 3, 4), (1, 2, 3, 5), and (1, 2, 3, 6). H is vertex-planar but not (Zykov-)planar. Figure 2(c) shows two drawings of H which showcase different methods to draw the hyperedges in a vertex-based Venn diagram. Subdivision Drawings. Vertex-based Venn diagrams are a particular type of hypergraph drawings which we call subdivision drawings. In a subdivision drawing each vertex corresponds uniquely to a face of a planar subdivision. Fur- thermore, for any hyperedge, the union of the faces corresponding to the vertices incident to that hyperedge, the hyperedge region, is connected. Hypergraph draw- ings which are based on Euler diagrams are subdivision drawings as well. Below we discuss them in some more detail. To draw a hypergraph H as an Euler diagram we again need to interpret H as a set system. Euler diagrams represent a collection of sets by simple, closed curves in the plane, such that the interior of each curve represents the elements of the corresponding set. Any face induced by the collection of curves is called a zone, which lies in the interior of some curves and in the exterior of the rest. In an Euler diagram, a zone z is only present if an element exists that is in exactly those sets whose curves have z as their common interior, and z is in the common exterior of all other curves [13]. No two zones may represent the same intersections of sets. Often, certain well-formedness conditions are considered part of the definition of Euler diagrams. These specify that no point may be the intersection of three of more curves, and all intersections of two curves are proper intersections (no two curves partially overlap). Flower and Howse call diagrams satisfying these conditions concrete Euler diagrams [8]. Extended Euler diagrams [16] allow curves to intersect in more general ways: They may partially coincide, and multiple intersections are allowed. Also, zones exist for all possible subsets that can be obtained by intersections. For exam- ple, the set system S : {{1, 2, 3}, {1, 2, 4}, {1, 3, 4}, {2, 3, 4}} generates all four units, all six pairs, and all four triples of elements as intersections of one or more sets, so an extended Euler diagram will have 14 zones for these intersections. Strictly speaking, an extended Euler diagram is not a subdivision drawing, since there is no unique mapping between the vertices and the faces of the subdivision anymore. (There is, however, an injective mapping.) There is no concrete Euler diagram for S, but if we interpret S as a hy- pergraph, then a vertex-based Venn diagram of S with four faces exists. On the other hand, hyperedge regions of a concrete Euler diagram are simply con- nected, whereas hyperedge regions of a vertex-based Venn diagram need only be connected, that is, they can have holes.