The Cartesian product of hypergraphs

The Cartesian Product of Hypergraphs Lydia Gringmann BIOINFORMATICS GROUP, DEPARTMENT OF COMPUTER SCIENCE AND INTERDISCIPLINARY CENTER FOR BIOINFORMATICS UNIVERSITY OF LEIPZIG, HÄRTELSTRASSE 16-18, D-04107 LEIPZIG, GERMANY E-MAIL: GLYDIA@BIOINF.UNI-LEIPZIG.DE Marc Hellmuth MAX PLANCK INSTITUTE FOR MATHEMATICS IN THE SCIENCES INSELSTRASSE 22, D-04103 LEIPZIG, GERMANY BIOINFORMATICS GROUP, DEPARTMENT OF COMPUTER SCIENCE AND INTERDISCIPLINARY CENTER FOR BIOINFORMATICS UNIVERSITY OF LEIPZIG, HÄRTELSTRASSE 16-18, D-04107 LEIPZIG, GERMANY E-MAIL: MARC@BIOINF.UNI-LEIPZIG.DE Peter F. Stadler BIOINFORMATICS GROUP, DEPARTMENT OF COMPUTER SCIENCE AND INTERDISCIPLINARY CENTER FOR BIOINFORMATICS UNIVERSITY OF LEIPZIG, HÄRTELSTRASSE 16-18, D-04107 LEIPZIG, GERMANY MAX PLANCK INSTITUTE FOR MATHEMATICS IN THE SCIENCES INSELSTRASSE 22, D-04103 LEIPZIG, GERMANY RNOMICS GROUP, FRAUNHOFER INSTITUT FÜR ZELLTHERAPIE UND IMMUNOLOGIE, DEUTSCHER PLATZ 5E, D-04103 LEIPZIG, GERMANY DEPARTMENT OF THEORETICAL CHEMISTRY, UNIVERSITY OF VIENNA, WÄHRINGERSTRAßE 17, A-1090 WIEN, AUSTRIA SANTA FE INSTITUTE, 1399 HYDE PARK RD., SANTA FE, NM87501, USA E-MAIL: STUDLA@BIOINF.UNI-LEIPZIG.DE 2 JOURNAL OF GRAPH THEORY ABSTRACT We show that every simple, (weakly) connected, possibly directed and infinite, hypergraph has a unique prime factor decomposition with respect to the (weak) Cartesian product, even if it has infinitely many factors. This generalizes previous results for graphs and undirected hypergraphs to directed and infinite hypergraphs. The proof adopts the strategy outlined by Imrich and Žerovnik for the case of graphs and introduced the notion of diagonal-free grids as a replacement of the chord-free 4-cycles that play a crucial role in the case of graphs. This leads to a generalization of relation δ on the arc set, whose convex hull is shown to coincide with the product relation of the prime factorization. Keywords: directed Hypergraph, Hypergraph, weak Cartesian Product, Prime Factor Decomposition, grid property 1. INTRODUCTION Directed hypergraphs are the common generalization of both directed graphs and (undirected) hypergraphs. A (directed) hypergraph H = (V, E) consists of a vertex set V and a family of (directed) hyperedges or hyperarcs E. Each hyperedge E ∈ E is an ordered pair of sets of vertices E = (T (E), H(E)) 6= (∅, ∅). The sets T (E) ⊆ V and H(E) ⊆ V are called the tail and head of E, respectively. To avoid the risk of confusion we will sometimes write V (H) and E(H) for the vertex set and the arc set of a hypergraph H. A hypergraph H = (V, E) is undirected if T (E) = H(E) for all E ∈ E. Most of the literature on hypergraphs is concerned with undirected hypergraphs [1], surveys on directed hypergraphs can be found in [7, 8]. Directed hypergraphs have been used as model for complex networks in biology and chemistry [15]. For instance a chemical reaction is naturally represented as a hyperedge E where T (E) lists the educts and H(E) the products of the chemical transformation. We say a hypergraph is finite if its vertex set and its edge set is finite. A hypergraph that is not finite is said to be infinite. A hypergraph H = (V, E) is called simple, if (i) |T (E) ∪ H(E)| > 1 for all E ∈ E and (ii) there are no two distinct arcs E, E ′ ∈ E such that T (E) = T (E ′ ) and H(E) = H(E ′ ). A hypergraph is a directed graph if |T (E)| = |H(E)| = 1 for all E ∈ E. We will be concerned with products of hypergraphs, more precisely with the most fundamental question that arises in this context: Does a hypergraph have a unique decomposition into prime factors? The answer to this question depends on the definition of the product. In the case of graphs, the product structures are quite well understood [11]. Several notions of hypergraph products have been studied in the literature, usually for restricted versions of undirected hypergraphs, see e.g. [2, 6, 9, 20]. The most commonly studied hypergraph product is the direct product, which consists of the Cartesian product of vertex sets, and the Cartesian (set) products of the hyperedges [3, 5, 18, 17]. For direct CARTESIAN PRODUCT OF HYPERGRAPHS 3 products of N -systems, i.e., directed hypergraphs satisfying |T (E)| = 1 and T (E) ⊆ H(E) for all E ∈ E, a prime factor theorem was proved in [13]. In this contribution we will focus on the Cartesian product of finite and infinite directed hypergraphs with finitely or infinitely many factors. The Cartesian graph product was introduced by Gert Sabidussi [19], who showed that connected graphs have a unique Cartesian prime factor decomposition. This result was generalized by Wilfried Imrich [9] to simple undirected hypergraphs: Definition. Let {Hi | i ∈ I} be a family of (finite or infinite), but undirected hypergraphs. The Cartesian product H = ✷i∈I Hi is defined as follows: V (H) = × V (Hi ) i∈I E(H) = {E ⊆ V (H) | pj (E) ∈ E(Hj ) for exactly one j ∈ I, and |pi (E)| = 1 for i 6= j}, where, for j ∈ I, pj : V (H) → V (Hj ) is the projection of the Cartesian product of the vertex sets into V (Hj ). The value pj (v) is also called j-th coordinate of vertex v. The Cartesian product of undirected hypergraphs is also considered for example in [3, 4, 5] A factorization algorithm for so-called conformal hypergraphs, a rather small class of finite and connected hypergraphs, with respect to the Cartesian product, is described in [4]. While the Cartesian product hypergraph of finitely many connected hypergraphs is connected, whether they are finite or not, this does not hold for the product of infinitely many hypergraphs: In this case, there are vertices that differ in infinitely many coordinates and hence are not connected by a path of finite length. As in the case of graphs [19], an infinite connected hypergraph that has infinitely many Cartesian prime factors cannot be the Cartesian product of its factors, but a connected component of this Cartesian product [10]. This gives rise to the notion of a weak Cartesian product: Definition. [10] Let {Hi | i ∈ I} be a family of hypergraphs and let ai ∈ V (Hi ) for i ∈ I. The weak Cartesian product H = ✷i∈I (Hi , ai ) of the “rooted” hypergraphs (Hi , ai ) is defined by V (H) = {v ∈ × V (Hi ) | pi(v) 6= ai for at most finitely many i ∈ I} i∈I E(H) = {E ⊆ V (H) | pj (E) ∈ E(Hj ) for exactly one j ∈ I, and |pi (E)| = 1 for i 6= j}. For finite index sets I, the weak Cartesian product does not depend on the choice of the ai and coincides with the usual Cartesian product. If I is infinite, it is the connected component of the Cartesian product containing a = (ai )i∈I . Every connected graph and undirected hypergraph has a unique representation as a weak Cartesian product [16, 10]. Here we extend and generalize these results further and show that every connected directed hypergraph has a unique prime factor decomposition with respect to the (weak) 4 JOURNAL OF GRAPH THEORY Cartesian product. Instead of following the proof strategies of the classical papers, we adopt the approach of Imrich and Žerovnik [14] that constructs a product relation σ starting from simpler relations on the edge set E. In the case of graphs, the Square Property [12] plays a central role as technical device. The Grid Property, which is introduced here, serves as generalization of this construction. Together with a generalization of the relation δ, we arrive at our main results: Theorem 1. Let γ be a convex equivalence relation on the arc set E(H) of a connected simple hypergraph H which satisfies the grid property. Then γ induces a factorization of H with respect to the weak Cartesian product. Theorem 2. Every connected simple hypergraph has a unique representation as a weak Cartesian product. Theorem 3. The product relation σ corresponding to the unique prime factor decomposition with respect to the weak Cartesian product of a connected simple hypergraph H equals the convex hull C(δ) of the relation δ. Since the Cartesian and the weak Cartesian product coincides for a finite number of factors, we also obtain the following corollaries. Corollary. The prime factor decomposition of a connected hypergraph with finitely many factors with respect to the Cartesian product is unique in the class of simple hypergraphs. Corollary. The prime factor decomposition of a connected hypergraph with respect to the Cartesian product is unique in the class of finite simple hypergraphs. 2. PRELIMINARIES As far as possible, we follow the notation and terminology of Berge’s classical book on hypergraphs [1], although our hypergraphs will in general be directed, T (E) 6= H(E) for some E ∈ E. For simplicity, we will refer to hyperarcs as arcs. Two arcs E, E ′ ∈ E(H) of a hypergraph H are incident if E ∩ E ′ 6= ∅, i.e., if there is a vertex that is contained in both arcs, independent of the directions. Two vertices x, y ∈ V are adjacent if there is an arc E containing them, i.e., x, y ∈ T (E) ∪ H(E), again without regard of direction. Concepts of sub-hypergraphs, paths, etc., are defined in the appendix to make this contribution self-consistent. For the set product V = ×i∈I Vi we define the projection pj : V → Vj by (v1 , v2 , . . . , vj , . . . ) 7→ vj . For subsets of V and ordered tuples of elements of V , the projection is defined element-wise. For example, for a hypergraph H = (×i∈I Vi , E) we CARTESIAN PRODUCT OF HYPERGRAPHS 5 have pj (E) = pj (T (E), H(E)) = (pj (T (E)), pj (H(E))) = ( [ v∈T (E) pj (v), [ pj (w)) . (1) w∈H(E) By abuse of notation, we will write |pj (E)| := |pj (T (E)) ∪ pj (T (H))| i.e., |pj (E)| refers to the cardinality of the union of the projections of head and tail of the arc E. Definition. Let Hi , i ∈ I be hypergraphs. The Cartesian product ✷i∈I Hi has the following vertex and arc sets: (1) V (✷i∈I Hi ) = ×i∈I V (Hi ), (2) and E = (T (E), H(E)) is an arc in ✷i∈I Hi if and only if there is a j ∈ I, such that (i) pj (E) = pj ((T (E), H(E))) = (T (pj (E)), H(pj (E))) ∈ E(Hj ) and (ii) |pi (E)| = 1 for all i 6= j. Figure 1 shows an example of a Cartesian product of two hypergraphs. H FIGURE 1. Hypergraph H and Cartesian product H✷E1,2 , where E1,2 is the hypergraph consisting of a single arc of three vertices such that |T (E1,2 )| = 1 and |H(E1,2 )| = 2. Lemma 1. The Cartesian Product H = ✷i∈I Hi of hypergraphs Hi is an undirected hypergraph if and only if all of its factors are undirected. Proof. The assertion follows directly from equ.(1). By construction, the Cartesian product is associative, commutative, distributive w.r.t. the disjoint union and has the trivial one vertex hypergraph K1 without arcs as unit. 6 JOURNAL OF GRAPH THEORY Lemma 2. The Cartesian product H = ✷ni=1 Hi of finitely many hypergraphs Hi is connected if and only if all of its factors Hi are connected. Proof. Because of associativity and commutativity it suffices to show the assertion for two factors, hence let H = H1 ✷H2 . First, assume that H1 and H2 are connected. Let v = (x, y) and v ′ = (x′ , y ′ ) be two arbitrary vertices in V (H). Consider a path Pxx′ = (E1 , . . . , Er ) from x to x′ in H1 and a path Pyy′ = (F1 , . . . , Fs ) from y to y ′ in H2 . Then (E1 × {y}, . . . , Er × {y}) is a path from (x, y) to (x′ , y) in H and ({x′ } × F1 , . . . , {x′ } × Fs ) is a path from (x′ , y) to (x′ , y ′ ) in H. Hence, (E1 × {y}, . . . , Er × {y}, {x′ } × F1 , . . . , {x′ } × Fs ) is a path from v to v ′ in H. W.l.o.g., suppose that H1 is not connected, i.e., it can be written in the form H1 = H1′ + H1′′ . Since the Cartesian product is distributive w.r.t. the disjoint union, we have H = H1′ ✷H2 + H1′′ ✷H2 . Hence H is the disjoint union of two hypergraphs, i.e., H is not connected. Let us now turn to the weak Cartesian product. As in the undirected case, the product of finitely many connected hypergraphs is connected, whether they are finite or not, but the product of infinitely many factors is not connected. Definition. Let {Hi | i ∈ I} be a family of hypergraphs and let ai ∈ V (Hi ) for i ∈ I. The weak Cartesian product H = ✷i∈I (Hi , ai ) of the hypergraphs (Hi , ai ) rooted at ai is given by V (H) ={v ∈ × V (Hi ) | pi(v) 6= ai for at most finitely many i ∈ I} i∈I E(H) ={E ⊆ V (H) | pj (E) = pj ((T (E), H(E))) = (T (pj (E)), H(pj (E))) ∈ E(Hj ) for exactly one j ∈ I, and |pi (E)| = 1 for i 6= j}. We will, in the following write ✷ai∈I Hi for ✷i∈I (Hi , ai ), where a ∈ V (H) such that pi (a) = ai for all i ∈ I. Again, the weak Cartesian product does not depend on the (ai )i∈I if I is finite. Moreover, it coincides with the Cartesian product if I is finite. The partial hypergraph of H induced by all vertices of H which differ from w ∈ V (H) exactly in the j-th coordinate is isomorphic to Hj . More formally h{v ∈ V (H) | pk (v) = wk for k 6= j}i ≃ Hj . We will call this partial hypergraph the Hj -layer through w and denote it by Hjw . The isomorphism Hjw → Hj is then the projection pj . For u ∈ V (Hjw ) we have Hju = Hjw and, moreover, V (Hju ) ∩ V (Hjw ) = ∅ if and only if u ∈ / V (Hjw ). Lemma 2 implies the following CARTESIAN PRODUCT OF HYPERGRAPHS 7 Corollary. A weak Cartesian product H = ✷ai∈I Hi is connected if and only if all of its factors Hi are connected. In this case, H = ✷ai∈I Hi is the connected component of the Cartesian product ✷i∈I Hi that contains the vertex a. Lemma 3. The Cartesian product H = ✷i∈I Hi of hypergraphs Hi is simple if and only if all of its factors Hi are simple. Proof. Let Exj denote an arc containing x ∈ V and whose projection on the j-th factor is an arc in the j-th factor. First let all Hi , i ∈ I be simple and suppose H is not simple. We have to examine two cases: First, suppose H contains at least one loop on vertex x with coordinates xi , i ∈ I. Hence there is a loop in some factor Hj , contradicting that Hj is simple. ′ Thus, it must hold |E| ≥ 2 for all E ∈ E(H). Second, let Exj and Eyk be two different ′ ′ ′ and j = k. arcs, such that T (Exj ) = T (Eyk ) and H(Exj ) = H(Eyk ). Hence, x ∈ Eyk ′ ′ Thus, pj (T (Exj )) = pk (T (Eyk )), as well as pj (H(Exj )) = pj (H(Eyj )). Therefore Hj is not simple, a contradiction. Now assume that (at least) one of the factors is not simple; w.l.o.g. say H1 . There are two possibilities: First, assume there is an arc E ∈ E(H1 ) with |E| = 1, say E = {x1 }. Hence, |E1x | = |{x}| = 1 for any x ∈ V (H) with p1 (x) = x1 and H would not be simple. Second, assume there are two different arcs Ei , Ej ∈ E(H1 ), such that T (Ei ) = T (Ej ) and H(Ei ) = H(Ej ). Then, the definition of the Cartesian product implies that there are two arcs E = Ei × {x} and E ′ = Ej × {x} in H with x ∈ ×l∈I\{1} V (Hl ) such that T (E) = T (E ′ ) and H(E) = H(E ′ ). Hence, H is not simple. Note that the weak Cartesian product ✷i∈I (Hi , ai ) is a partial hypergraph of the Cartesian product ✷i∈I Hi , that is induced by V (✷i∈I (Hi , ai )). Hence we have Corollary. The weak Cartesian product H = ✷i∈I (Hi , ai ) of hypergraphs Hi is simple if and only if all of its factors Hi are simple. From here on we assume that all hypergraphs are simple. 3. UNIQUE PRIME FACTOR DECOMPOSITION A hypergraph H is prime with respect to the (weak) Cartesian product if it cannot be represented as the (weak) Cartesian product of two nontrivial hypergraphs. A prime factor decomposition (PFD) of H is a representation as a Cartesian product H = ✷i∈I Hi , or as a a weak Cartesian product H = ✷ai∈I Hi , resp., such that all factors Hi , i ∈ I, are prime and Hi 6≃ K1 . In order to show that every hypergraph has a unique representation as a weak Cartesian product, we follow the strategy of Imrich and Žerovnik [14] and characterize the socalled product relations defined on the arc set E of H. The advantage of this approach is 8 JOURNAL OF GRAPH THEORY that it does not require finiteness and hence also pertains to the weak Cartesian product of infinitely many factors. Definition. A product relation is an equivalence relation on the arc set E(H) of a (weak) Cartesian product H = ✷ai∈I Hi of (not necessarily prime) hypergraphs Hi such that, for E, F ∈ E(H), E and F are in relation γ, EγF , if and only if there exists a j ∈ I, such that |pj (E)| > 1 and |pj (F )| > 1, and |pi (E)| = |pi (F )| = 1 holds for all i 6= j. If all factors Hi are prime, we denote this relation by σ. Note that E and F are in relation σ if and only if their vertices differ in the same coordinate w.r.t. to the PFD. Let Σi , i ∈ I be the equivalence classes of σ. By construction, every connected component of a partial hypergraph generated by the arcs of an equivalence class Σi is isomorphic S to Hi . Furthermore, every union of σ-equivalence classes j∈J Σj , J ⊆ I generates a partial hypergraph of H, whose connected components are isomorphic to HJ := ✷aj∈J Hj . THE GRID PROPERTY Definition. Let G be a collection of arcs of H of the form Ea = (T (Ea ), H(Ea )), a ∈ A and Fb = (T (Fb ), H(Fb )), b ∈ B. We say that G is an |A| × |B|-grid if, for all a, a′ ∈ A and b, b′ ∈ B with a 6= a′ and b 6= b′ the following two conditions are satisfied: (i) Ea and Fb have exactly one vertex in common, i.e., (T (Ea ) ∪ H(Ea )) ∩ (T (Fb ) ∪ H(Fb )) = {zab }, and (ia) zab ∈ T (Fb ) (resp. H(Fb )) if and only if zab′ ∈ T (Fb′ ) (resp. H(Fb′ )) for all b′ ∈ B, and (ib) zab ∈ T (Ea ) (resp. H(Ea )) if and only if za′ b ∈ T (Ea′ ) (resp. H(Ea′ )) for all a′ ∈ A, and (ii) Ea and Ea′ have no common vertex for a 6= a′ , i.e., (T (Ea ) ∪ H(Ea )) ∩ (T (Ea′ ) ∪ H(Ea′ )) = ∅. Analogously, Fb and Fb′ are disjoint for b 6= b′ . An arc D ∈ E(H) satisfying zab ∈ D and za′ b′ ∈ D for all for a, a′ ∈ A and b, b′ ∈ B with a 6= a′ and b 6= b′ , is a diagonal of the |A| × |B|-grid G. The construction of the grid implies that Ea and Fb satisfy |Ea | = |B| and |Fb | = |A| for all a ∈ A, and b ∈ B. Diagonal-free grids appear whenever two arcs of two hypergraphs are multiplied with respect to the Cartesian product. In this sense they generalize the chordless squares appearing as Cartesian products of arcs of undirected simple graphs. This suggests to generalize the relation δ [19, 14] in the following way: Definition. Let H be a connected hypergraph. For E, F ∈ E(H) we say E and F are in relation δ, EδF , if one of the following conditions is satisfied: CARTESIAN PRODUCT OF HYPERGRAPHS 9 (i) E and F have no vertex in common and form the opposite arcs of a 4-cycle. (ii) E and F are incident to at least one common vertex and there is no grid without diagonals that contains both E and F . (iii) E = F . Note that whenever E and F share two or more vertices there is no (|E| × |F |)-grid that contains E and F , and hence EδF . Obviously, the δ is reflexive and symmetric. Its transitive closure δ ∗ , i.e., the smallest transitive relation containing δ, is therefore an equivalence relation. Condition (ii) implies that any two incident arcs E and F with E /δ F span an (|E| × |F |)-grid without diagonals, whose arcs we denote by E, {Ea }a∈A and F , {Fb }b∈B , where EδEa and F δFb for all a ∈ A and b ∈ B, respectively. Lemma 4. Two incident arcs that are not in relation δ span a unique (|E| × |F |)-grid. Proof. Suppose there exists another (|E| × |F |)-grid consisting of arcs E, {Ea′ }a∈A and F , {Fb′ }b∈B . Then there must be a k ∈ A and an l ∈ B such that Ek′ ∈ / {Ea }a∈A and Fl′ ∈ / {Fb }b∈B , respectively. Hence there exists both an arc Er ∈ {Ea }a∈A and and arc Fs ∈ {Fb }b∈B such that Ek′ and Er have common vertices and Fl′ and Fs have common vertices. Thus there is a 4-cycle Ek′ Er Fs Fl′ , where Ek′ and Fs as well as Er and Fl′ are opposite arcs. Hence Ek′ δFs and Er δFl′ , and therefore (E, F ) ∈ δ ∗ . It follows that if E and F belong to distinct δ ∗ -equivalence classes, they span a unique (|E| × |F |)-grid. This observation suggests the following definition: Definition. Let γ be an equivalence relation on the arc set E(H) of a hypergraph H. We say γ has the grid property if any two adjacent arcs E and F of H with E γ/ F span exactly one diagonal-free |E| × |F |-grid. Our discussion above implies that δ ∗ has the grid property. Let γ be an arbitrary equivalence relation on the arc set of a hypergraph H that contains δ ∗ . For any two arcs E and F with E γ/ F we also have E /δ ∗ F and therefore, they span exactly one (diagonal-free) |E| × |F |-grid. As a consequence, every equivalence relation γ that contains δ ∗ satisfies the grid property. Lemma 5. Let H be a connected hypergraph and let γ be an equivalence relation on E(H) satisfying the grid property. Denote the equivalence classes of γ by Γi , i ∈ I. Then every vertex of v ∈ V (H) is incident to an arc E ∈ Γi for every i ∈ I. Proof. Suppose that there is an equivalence class Γi of γ and a set of vertices that is not contained in any Γi -arc. By connectedness of H, there is a pair of vertices u, v ∈ V (H) and an arc E ∈ E(H) with {u, v} ⊆ T (E) ∪ H(E) such that u belongs to a Γi -arc, say F , and there is no Γi -arc containing v. It follows that E ∈ / Γi , i.e., it must be contained in some other equivalence class Γk , k 6= i. By construction, E and F are two incident arcs 10 JOURNAL OF GRAPH THEORY belonging to different equivalence classes of γ and hence span a grid. Thus there must be a Γi -arc containing v, contradicting the assumption. Lemma 6. Let H = ✷ai∈I Hi be a weak Cartesian product of prime hypergraphs Hi and let E, F ∈ E(H). If E and F are in relation δ, they are in relation σ. Proof. Suppose first, that for the arcs E and F holds E ∩ F = ∅ and there are arcs E ′ , F ′ ∈ E(H) such that {E, E ′ , F, F ′ } is a 4-cycle. Moreover, we denote with c(E) the coordinates where the vertices of the arc E differ. W.l.o.g. assume x1 is common to E and E ′ , x2 is common to E and F ′ , y1 is common to F and E ′ , and y2 is common to F and F ′ . The coordinates varying within these arcs are denote by c(E) = i, c(F ) = j, c(E ′ ) = i′ , c(F ′ ) = j ′ ; see Fig. 2. c(E) = i xr x1 c(F ′ ) = j ′ c(E ′ ) = i′ ys y1 c(F ) = j FIGURE 2. 4-cycle EE ′ F F ′ Then we have: pk (x1 ) = pk (x2 ) pk (y1 ) = pk (y2 ) for all k 6= i for all k 6= j (2) (3) pk (x1 ) = pk (y1 ) pk (x2 ) = pk (y2 ) for all k 6= i′ for all k 6= j ′ . (4) (5) pk (x1 ) = pk (y2 ) for all k 6= i, j ′ (6) pk (x1 ) = pk (y2 ) for all k 6= j, i′ . (7) It follows from (2) and (5) that and from (3) and (4) Therefore we have either i = j and i′ = j ′ or i = i′ and j ′ = j. Assume i 6= j. Then the latter case must hold and we have pk (x2 ) = pk (x1 ) = pk (y2 ) for all k 6= i, and since CARTESIAN PRODUCT OF HYPERGRAPHS 11 x2 6= y1 holds, pi (x2 ) 6= pi (y1 ) = pi (y2 ) and pj (x2 ) 6= pj (y2 ). Hence, x2 and y2 differ in more than one coordinate, thus they cannot lie in the same arc F ′ , which contradicts the assumption, therefore, i = j must hold, i.e. c(E) = c(F ), and therefore EσF . Now let E and F be incident arcs of a hypergraph H and assume that there is no |E| × |F |-grid without diagonals containing them. First, consider the case that E and F share more than a single vertex. Then there is an index i ∈ I such that |pi (E)| > 1 and in particular, |pi (E ′ )| > 1 holds for all E ′ ⊆ T (E) ∪ H(E) with |E ′ | > 1. Since E and F have more than one vertex in common it follows that |pi ((T (E) ∪ H(E)) ∩ (T (F ) ∪ H(F ))| > 1 and hence |pi (E)| > 1 and |pi (F )| > 1, and thus EσF . Now suppose that E and F share a single vertex S v and assume that E and F are not in S relation σ. Let Ê = T (E) ∪ H(E) = {v} ∪ a∈A {xa } and F̂ = T (F ) ∪ H(F ) = {v} ∪ b∈B {yb }. Furthermore set j = c(F ) and observe that j 6= i. For all xa ∈ E, a ∈ A, and all b ∈ B there exist vertices zab ∈ V (H) such that pi (zab ) = pi (xa ) pk (zab ) = pk (yb ) (8) for all k 6= i. (9) Using (9) and the fact, that S {v, yb } ⊆ F for all b ∈ B and {v, xa } ⊆ E, we can conclude that the set F̂a = {xa } ∪ b∈B {zab } satisfies pj (F̂a ) = {pj (xa )} ∪ [ {pj (zab )} = {pj (v)} ∪ b∈B [ {pj (yb )} = pj (F̂ ) (10) b∈B as well as pk (zab ) = pk (yb ) = pk (v) = pk (xa ) for all k 6= i, j and for all b ∈ B (11) Now we use (8) and (11) and obtain for all k 6= i and for all b ∈ B. pk (zab ) = pk (xa ) Analgously, the set Êb = {yb } ∪ pi (Êb ) = {pi (yb )} ∪ [ a∈A S a∈A {zab } (12) satisfies {pi (zab )} = {pi (v)} ∪ [ {pi (xa )} = pi (Ê). a∈A Again, we use (8) and the fact that {v, yb } ⊆ F for all b ∈ B. (13) 12 JOURNAL OF GRAPH THEORY T (F ) T (E) v T (Fa ) x1 y1 T (Eb ) FIGURE 3. yb xa xr H(E) za1 z1b zab ys zas H(F ) H(Fa ) zrb H(Eb ) Arcs E and F (thick arcs) with (E, F ) ∈ / σ and the |E| × |F |-grid they span It will be useful to relabel v as z00 , Ê as Ê0 , F̂ as F̂0 , the xl as zl0 for all l ∈ A, and the yl as z0l for all l ∈ B. Then we have Êb ∩ F̂a = {zab } (14) for all a ∈ A0 := A ∪ {0} and b ∈ B0 := B ∪ {0}. Since z0b ∈ T (F ) (resp. H(F )) if and only if pj (z0b ) ∈ pj (T (F )) (resp. pj (H(F ))) for all b ∈ B0 , we can conclude by the definition of the (weak) Cartesian product and from equations (10) and (12) that for all F̂a , a ∈ A, there exists an arc Fa = (T (Fa ), H(Fa )) with |Fa | = |F | such that F̂a = T (Fa ) ∪ H(Fa ). More precisely, we have T (Fa ) = {zab | b ∈ B0 and z0b ∈ T (F )}, resp. H(Fa ) = {zab | b ∈ B0 and z0b ∈ H(F )} (15) for all a ∈ A. An analogous argument can be made for Êb : Since za0 ∈ T (E) (resp. H(E)) if and only if pi (za0 ) ∈ pi (T (E)) (resp. pi (H(E))) for all a ∈ A0 , it follows from equations (9) and (13) that for all Êb , b ∈ B, there exists an arc Eb = (T (Eb ), H(Eb )) with |Eb | = |E| such that Êb = T (Eb ) ∪ H(Eb ). More precisely, we have T (Eb ) = {zab | a ∈ A0 and za0 ∈ T (E)}, resp. H(Eb ) = {zab | a ∈ A0 and za0 ∈ H(E)} for all b ∈ B. (16) CARTESIAN PRODUCT OF HYPERGRAPHS 13 Equation (14) immediately implies that the intersection of more then two arcs is empty. Consequently, whenever two adjacent arcs E and F are not in relation σ, they span an |E| × |F |-grid with |E| × |F | vertices. It remains to show that these grids have no diagonals. Therefore, we need to show that there is no D ∈ E(H) such that {zab , za′ b′ } ⊆ D for all a, a′ ∈ {0} ∪ A, b, b′ ∈ {0} ∪ B with a 6= a′ and b 6= b′ . For zab , za′ b′ we have: (8) (8) (9) (9) pi (zab ) = pi (za0 ) 6= pi (za′ 0 ) = pi (za′ b′ ) (17) pj (zab ) = pj (z0b ) 6= pj (z0b′ ) = pj (za′ b′ ) (18) The inequalities follow from the fact, that {za0 , za′ 0 } ⊆ E0 and {z0b , z0b′ } ⊆ F0 . That means that zab and za′ b′ differ in more than one coordinate, hence, they cannot be contained in the same arc, which completes the proof. Lemma 6 implies that δ ⊆ σ and since σ is an equivalence relation, δ ∗ ⊆ σ holds as well. Thus, the product relation σ has the grid property. CONVEXITY Lemma 7. Let H = ✷ai∈I Hi be a weak Cartesian product of connected hypergraphs Hi . Then each HJ = ✷aj∈J Hj -layer is convex for any index set J ⊆ I. Proof. It suffices to show that, whenever there is a path P between two arbitrary vertices u and v of the same HJ -layer HJu containing no arcs of this layer, then there exists a path Q which entirely lies in HJu and satisfies |Q| < |P |. Suppose P = (u = u0 , E1 , u1 , E2 , . . . , uk−1 , Ek , uk = v). Since u and v belong to the same Hj -layer, pl (u) = pl (v) holds for all l ∈ I \ J. There must be an arc Ei of P such that Ei is contained in some HJ layer, by assumption different from HJu . Otherwise we would have pl (u) = pl (v) for all l ∈ J, hence pl (u) = pl (v) for all l ∈ I, i.e. u = v. Let {Ej1 , . . . , Ejr } be a subset of arcs of P , with j1 , j2 , . . . , jr ∈ {1, . . . , k}, j1 < j2 < . . . < jr , that are in some HJ -layer different from HJu , and no arc is the copy of another. To be more precise, for each ji there is a ki ∈ J with pki (Eji ) ∈ E(Hki ) and pka (Eja ) 6= pkb (Ejb ) and jr is maximal. Note that a 6= b does not imply ka 6= kb . for a 6= b (19) 14 JOURNAL OF GRAPH THEORY u E1 vi−1 F1 Fr v vi v1 vr−1 Fi HJu u1 Ejr Ej1 j1 − 1 j1 jr − 1 ji − 1 jr HJj1 HJji ji Eji FIGURE 4. Idea of the proof: Path P and Path Q (thick) which we got by shifting the arcs Eji in the HJu -layer By assumption, E1 is not contained in any HJ -layer, thus r < k. Without loss of generality, we can assume that the Eji are not incident. In the following we will denote the vertices uji by ji . If we set j0 := u0 , we have pl (ji−1 ) = pl (ji − 1) for all l ∈ J and all i ∈ {1, . . . , r} (20) and since ji − 1, ji ∈ Eji , pl (ji ) = pl (ji − 1) (21) holds for all l 6= ki , ki ∈ J. Furthermore, for each ji , i ∈ {1, . . . , r}, there exists a vi ∈ V (H), such that pl (vi ) = pl (u) for all l ∈ I \ J pl (vi ) = pl (ji ) for all l ∈ J (22) (23) In particular, Equation (22) implies vi ∈ V (HJu ) for all i ∈ {1, . . . , r}. It follows (23) pl (vi−1 ) = pl (ji−1 ) (20),(21) = (23) pl (ji ) = pl (vi ) for all l ∈ J \ {ki } (24) and by Equation (22) we have pl (vi−1 ) = pl (vi ) for all l ∈ I \ J (25) for all l 6= ki (26) hence by Equations (24) and (25) pl (vi−1 ) = pl (vi ) CARTESIAN PRODUCT OF HYPERGRAPHS 15 In other words, any two vertices vi−1 , vi lie in the same Hki layer for some ki ∈ J. Next we show that there are arcs Fi ∈ E(HJu ) containing both vi−1 and vi . From Equations (20) and (23) it follows (20) (23) pki (vi−1 ) = pki (ji−1 ) = pki (ji − 1) (23) pki (vi ) = pki (ji ) (27) (28) Thus we have by Equations (27), (28) and (19) pki (vi−1 ), pki (vi ) ∈ pki (Eji ) (29) Hence by Equations (22), (26) and (29), for each i ∈ {1, . . . , r} there exists an arc Fi in HJu , such that vi−1 , vi ∈ Fi . Now consider the vertex vr . Since there is no more arc Ej , j > jr , of P that is cJ := ✷i∈I\J Hi -layer Therefore contained in any HJ -layer, jr and v belong to the same H we have (23) pl (vr ) = pl (jr ) = pl (v) for all l ∈ J (30) and from the definition of vr and the fact that u and v are in the same HJ -layer, it follows (22) pl (vr ) = pl (v) for all l ∈ I \ J. (31) Therefore, we can conclude vr = v, and we have found a path Q = (u = v0 , F1 , v1 , . . . , vr−1 , Fr , vr = v) from u to v, whose arcs all lie entirely within HJu . Furthermore, we have |Q| = r < k = |P |, which completes the proof. Definition. An equivalence relation γ on the arc set E(H) of a connected hypergraph H with equivalence classes Γi , i ∈ I, is called convex S if for any J ⊆ I every connected component of the partial hypergraph generated by i∈J Γi is convex. By Lemma 7, the product relation σ is a convex relation. Moreover, any product relation must be convex and has to satisfy the grid property. Lemma 8. Let γ be an equivalence relation on the arc set E(H) of a connected hypergraph H which satisfies the grid property. Let Γ be an equivalence class of γ. If all connected components of the partial hypergraph of H generated by Γ are convex, they are isomorphic. Proof. Let HΓ be the partial hypergraph generated by Γ with connected components b denote the union of all equivalence classes of γ, distinct from Γ, i.e., Ci , i ∈ I and let Γ 16 JOURNAL OF GRAPH THEORY b = S ′ Γ′ . It suffices to show that any two components C1 and C2 that are connected Γ Γ 6=Γ b by a Γ-arc are isomorphic. We define a mapping ϕ : V (C1 ) → V (C2 ) such that x 7→ ϕx, b whenever x and ϕx are connected by a Γ-arc. From the grid property and Lemma 5, it follows that for all x ∈ V (C1 ) there exists a ϕx ∈ V (C2 ). The grid property ensures that adjacent vertices in C1 have different images in C2 and arcs in C1 map onto arcs in C2 , such that ϕ(E) = ϕ((T (E), H(E))) = (T (ϕ(E)), H(ϕ(E))) ∈ E(C2 ) for all E ∈ E(C1 ). Convexity implies that non adjacent vertices in C1 have different images in C2 as well, i.e., the mapping ϕ is injective. On the other hand we can extend ϕ−1 to a mapping ψ : V (C2 ) → V (C1 ). Analogously, it follows that for all y ∈ V (C2 ) there is a ψy in V (C1 ), hence ϕ−1 = ψ, i.e., ϕ is bijective, and every arc in C2 maps onto an arc in C1 , thus ϕ is an isomorphism between C1 and C2 . Sometimes, the transitive closure of δ is already convex. If this is the case, then each path between two vertices of the same connected component of an equivalence class of δ ∗ must contain at least one arc of this equivalence class as a consequence of Lemma 7. Lemma 9. Let γ be an equivalence relation on the arc set E(H) of a connected hyperb Let HΓ graph H satisfying the grid property with only two equivalence classes Γ and Γ. b with connected components and HΓb be the partial hypergraphs generated by Γ and Γ, bj , j ∈ J, respectively. Then Ci , i ∈ I, and C bj ) 6= ∅ V (Ci ) ∩ V (C for all i ∈ I, j ∈ J. In particular, bj )| = 1 |V (Ci ) ∩ V (C bj are convex. holds if Ci and C bj with V (Ci ) ∩ V (C bj ) = ∅, such that they Proof. Suppose there are components Ci , C have minimal distance. Let P = (v0 , E1 , v1 , E2 , . . . , Ek , vk ) be a shortest path from Ci to bj , such that v0 ∈ V (Ci ) and vk ∈ V (C bj ). Obviously, the first arc E1 must lie in Γ b and C the vertex v1 is not in Ci , otherwise E1 would be in Γ which contradicts the minimality of P . Lemma 5 implies that v1 must be contained in a Γ-component, say Ck . Since the bj is smaller than |P |, we have V (Ck ) ∩ V (C bj ) 6= ∅. Let w be a distance from Ck to C ′ bj ) and let P be a path from v1 to w in Ck . By repeated application vertex in V (Ck )∩V (C b of the grid property we obtain a vertex u in V (Ci ) connected to w by a Γ-arc. But then b b u must be in V (Cj ) and thus |V (Ci ) ∩ V (Cj )| ≥ 1. bj )| ≥ 2. Let u, w ∈ V (Ci ) ∩ V (C bj ). By connectivity we Now assume |V (Ci ) ∩ V (C ′ bj as well. Therefore have a path Q from u to w in Ci and a path Q from u to w in C CARTESIAN PRODUCT OF HYPERGRAPHS 17 bj or both are not either |Q| > |Q′ | or |Q′ | > |Q| or |Q| = |Q′ | holds. Hence either Ci or C convex, and thus the second assertion holds. PROOF OF THE THEOREMS We are now able to prove Theorem 1: Proof of Theorem 1. First assume that γ has only two equivalence classes Γ and b with connected components Ci , i ∈ J and C bj , j ∈ Jb respectively, of the generated Γ partial hypergraphs. By Lemma 9 we can assign uniquely determined coordinates (i, j) bj ), i ∈ J, j ∈ J. b On the other to each vertex of H, whenever {v} = V (Ci ) ∩ V (C hand for all such coordinates there exists a uniquely determined vertex in V (H), since bj )| = 1. |V (Ci ) ∩ V (C In the following we will identify each vertex of H with its coordinates. Obviously we b for all i ∈ J and V (C bj ) = {(i, j) | i ∈ J} for all j ∈ J. b have V (Ci ) = {(i, j) | j ∈ J} Recall that Lemma 8 implies that the Ci are isomorphic for all i ∈ J. In particular C1 ≃ Ci holds for all i ∈ J. The isomorphism is given by the mapping b (1, j) 7→ (i, j) for all j ∈ J. If C1 and Ci are connected by an arc, it is an isomorphism as in the proof of Lemma 8. If they are connected by a path, it is an isomorphism by induction on the length of the b1 ≃ C bj for all j ∈ Jb given by the isomorphism path. Analogously we have C (i, 1) 7→ (i, j) for all i ∈ J. A set of vertices E = {(i1 , j1 ), . . . , (iq , jq ), . . .}, i1 , . . . , iq , . . . ∈ J, j1 , . . . , jq , . . . ∈ b is an arc in H with T (E) = {(it1 , jt1 ), . . . , (itr , jtr ), . . .} and H(E) = J, {(ih1 , jh1 ), . . . , (ihs , jhs ), . . .} if and only if either (i) it is in the same Ci , hence i1 = . . . = iq = . . . = i and E1 = {(1, j1 ), . . . , (1, jq ), . . .} is an arc in C1 , such that T (E1 ) = {(1, jt1 ), . . . , (1, jtr ), . . .} and H(E1 ) = {(1, jh1 ), . . . , (1, jhs ), . . .}, or bj , hence j1 = . . . = jq = . . . = j and E2 = {(i1 , 1), . . . , (iq , 1)} (ii) it is in the same C b is an arc in C1 , such that T (E2 ) = {(it1 , 1), . . . , (itr , 1), . . .} and H(E2 ) = {(ih1 , 1), . . . , (ihs , 1), . . .}. b1 . That is, H is isomorphic to C1 ✷C Now define hypergraphs H1 and H2 by setting V (H1 ) = {i : (i, 1) ∈ V (C1 )} and b1 )}. H1 and H2 are isomorphic to C1 and C b1 by the isomorphic V (H2 ) = {j : (1, j) ∈ V (C mappings i 7→ (i, 1) and j 7→ (1, j) respectively, thus H = H1 ✷H2 . Assume now that γ has arbitrarily many equivalence classes Γi , i ∈S I. Let γi be bi = the equivalence relation with the two equivalence classes Γi and Γ k∈I,k6=i Γk for 18 JOURNAL OF GRAPH THEORY bi arbitrary i ∈ I. As already shown, we get a factorization of H into two factors Hi ✷H b b where Hi and Hi belong to Γi and Γi , respectively. We will call the projection, more b i into the factor Hi the i-th precisely the image of the projection, of a vertex v in Hi ✷H i coordinate of v, denoted by v . Clearly, we can assign coordinates to each vertex. If two vertices u, v have the same i-th coordinate, then, by convexity, there is no Γi -arc on any shortest path between them. Thus, if u and v have the same coordinates, there is no nontrivial shortest path between them, hence u = v. It follows that the assignment of coordinates to vertices of a connected hypergraph H is bijective. A subset E = {v1 , . . . , vq , . . .} of V (H) is an arc of H with T (E) = {vt1 , . . . , vtr , . . .} and H(E) = {vh1 , . . . , vhs , . . .} if and only if the vl differ in the same coordinate, say the i-th, for all l ∈ {1, . . . , q, . . .} and Ei = {v1i , . . . , vqi , . . .} is an arc in Hi with T (Ei ) = {vti1 , . . . , vtir , . . .} and H(Ei ) = {vhi 1 , . . . , vhi s , . . .}. Since H is connected, its vertices differ in at most finitely many coordinates, thus we have H ≃ ✷ai∈I Hi for any a ∈ V (H). The equivalence relation whose only equivalence class is the entire arc set of a hypergraph H is trivially convex and satisfies the grid property and is therefore a product relation. This relation always exists. By Theorem 1 we can conclude that any convex relation on the arc set of a connected hypergraph that satisfies the grid property is a product relation and induces a factorization of this hypergraph. The smallest convex relation satisfying the grid-property, if such a relation exists at all, therefore must induce a PFD with respect to the Cartesian product. The next lemma is well known for undirected graphs. Its proof is one of the key application of the square-property [14]. This proof step can directly be generalized to the grid-property in the case of hypergraphs. Moreover, since the definition of a path (i.e., a weak path) coincides with the definition of a path in the undirected case, we can immediately state the next lemma for hypergraphs. Lemma 10. Let γj , j ∈ J be an arbitraryTset of convex relations on the arc set E(H) of a hypergraph H containing δ. Then γ = j∈J γj is convex. It is clear that for arbitrary equivalence relations on the arc set of a hypergraph, which satisfy the grid-property, their intersection also has the grid-property. Therefore Lemma 10 implies that there is exactly one finest convex equivalence relation on the arc set E(H) of a hypergraph H satisfying the grid property, namely the intersection of all convex relations on E(H) containing δ, that is its convex hull, C(δ). Conversely, any product relation must be convex and contains δ. Thus we have proved the Theorems 2 and 3. We conclude, furthermore Corollary. The PFD of a connected hypergraph with respect to the Cartesian product is unique in the class of finite simple hypergraphs. CARTESIAN PRODUCT OF HYPERGRAPHS 19 ACKNOWLEDGMENTS We thank Wilfried Imrich for many stimulating discussions on graph and hypergraph products. References [1] C. Berge. Hypergraphs: Combinatorics of finite sets. North-Holland, Amsterdam, NL, 1989. [2] C. Berge and M. Simonovitis. The coloring numbers of the direct product of two hypergraphs. In Hypergraph Seminar, volume 411 of Lecture Notes in Mathematics, pages 21–33, Berlin, Heidelberg, 1974. Springer-Verlag. [3] A. Bretto. Hypergraphs and the helly property. Ars Comb., 78:23–32, 2006. [4] A. Bretto, Y. Silvestre, and T. Vallée. Cartesian product of hypergraphs: properties and algorithms. In 4th Athens Colloquium on Algorithms and Complexity (ACAC 2009), volume 4 of EPTCS, pages 22–28, 2009. [5] W Dörfler. Multiple Covers of Hypergraphs. Annals N.Y. Acad. Sci., 319:169–176, 1979. [6] W. Dörfler and D. A. Waller. A category-theoretical approach to hypergraphs. Arch. Math., 34:185–192, 1980. [7] G. Gallo, G. Longo, S. Pallottino, and S. Nguyen. Directed hypergraphs and applications. Discrete Appl. Math., 42:177–201, 1993. [8] G. Gallo and M. G. Scutellà. Directed hypergraphs as a modelling paradigm. Rivista AMASES, 21:97–123, 1998. [9] W. Imrich. Kartesisches Produkt von Mengensystemen und Graphen. Studia Sci. Math. Hungar., 2:285–290, 1967. [10] W. Imrich. über das schwache Kartesische Produkt von Graphen. J. Comb. Theory, 11:1– 16, 1971. [11] W. Imrich and S. Klavžar. Product Graphs: Structure and Recognition. John Wiley & Sons, New York, 2000. [12] W. Imrich, S. Klavžar, and D. F. Rall. Graphs and Their Cartesian Product. Topics in Graph Theory. A K Peters, Ltd., Wellesley, MA, 2008. [13] W. Imrich and P. F. Stadler. A prime factor theorem for a generalized direct product. Discussiones Math. Graph Th., 26:135–140, 2006. [14] W. Imrich and J. Žerovnik. Factoring Cartesian-product graphs. J. Graph Theory, 18:557– 567, 1994. [15] S Klamt, U-U Haus, and F Theis. Hypergraphs and cellular networks. PLoS Comput Biol, 5:e1000385, 2009. [16] D. J. Miller. Weak cartesian product of graphs. Colloq. Math., 21:55–74, 1970. [17] D. Mubayi and V. Rödl. On the chromatic number and independence number of hypergraph products. J. Comb. Theory, Ser. B, 97:151–155, 2007. 20 JOURNAL OF GRAPH THEORY [18] R. Pemantle, J. Propp, and D. Ullman. On tensor powers of integer programs. SIAM J. Discrete Math., 5:127–143, 1992. [19] G. Sabidussi. Graph multiplication. Math. Z., 72:446–457, 1960. [20] X. Zhu. On the chromatic number of the product of hypergraphs. Ars Comb., 34:25–31, 1992. APPENDIX: BASIC DEFINITIONS A partial hypergraph H ′ ⊆ H of a hypergraph H = (V, E) is a hypergraph H ′ = (V ′ , E ′ ) with V ′ ⊆ V and E ′ ⊆ E. A partial hypergraph H ′ = (V ′ , E ′ ) ⊆ H = (V, E) is generated by the arc set E ′ if V ′ = ∪E∈E ′ E. It is induced by the vertex set V ′ , H ′ = hV ′ i, if E ′ = (E ∈ E | E ⊆ V ′ ). A strong path or simple path P of length k in H = (V, E) is a sequence P = [v0 , E1 , v1 , E2 , . . . , Ek , vk ] of distinct vertices and arcs of H, such that v0 ∈ T (E1 ), vk ∈ H(Ek ) and vj ∈ H(Ej ) ∩ T (Ej+1 ). A weak path P w of length k in a hypergraph H = (V, E) is a sequence P w = (v0 , E1 , v1 , E2 , . . . , Ek , vk ) of distinct vertices and arcs of H, such that v0 ∈ T (E1 ) ∪ H(E1 ), vk ∈ T (Ek ) ∪ H(Ek ) and vj ∈ [T (Ej ) ∪ H(Ej )] ∩ [T (Ej+1 ) ∪ H(Ej+1 )]. For the sake of convenience, we will refer to weak paths simply as paths. A hypergraph H is said to be weakly connected or simply connected for short, if each two vertices of H can be connected by a path. A path between two partial hypergraphs H ′ and H ′′ of a hypergraph H is a path in H between two vertices v ∈ V (H ′ ) and w ∈ V (H ′′ ). The distance between two vertices of a hypergraph is the length of the shortest path connecting them. The distance dH (H ′ , H ′′ ) between two partial hypergraphs H ′ , H ′′ is the minimal length of a path between H ′ and H ′′ . A partial hypergraph H ′ ⊆ H is convex if all shortest paths of H between vertices of H ′ are contained in H ′ . A 4-cycle in H is a partial hypergraph generated by arcs E1 , E2 , E3 , and E4 , such that Ei ∩ Ei+1 6= ∅, where the subscripts are taken modulo 4. A homomorphism from H1 = (V1 , E1 ) into H2 = (V2 , E2 ) is a mapping ϕ : V1 → V2 such that ϕ(E) is an arc in H2 whenever E is an arc in H1 with the property that ϕ(T (E)) = T (ϕ(E)) and ϕ(H(E)) = H(ϕ(E)). A mapping ϕ : V1 → V2 is a weak homomorphism if arcs are mapped either on arcs or on vertices. A bijective homomorphism ϕ whose inverse function is also a homomorphism is called an isomorphism.