Discussiones Mathematicae Graph Theory 40 (2020) 279–295 doi:10.7151/dmgt.2138 NICHE HYPERGRAPHS OF PRODUCTS OF DIGRAPHS Martin Sonntag Faculty of Mathematics and Computer Science TU Bergakademie Freiberg Pr¨uferstraße 1, D–09596 Freiberg, Germany e-mail: [email protected] and Hanns-Martin Teichert Institute of Mathematics University of L¨ubeck Ratzeburger Allee 160, D–23562 L¨ubeck, Germany e-mail: [email protected] Abstract If D = (V, A) is a digraph, its niche hypergraph NH(D)=(V, E) has − + the edge set E = {e ⊆ V | |e| ≥ 2 ∧∃ v ∈ V : e = ND (v) ∨ e = ND (v)}. Niche hypergraphs generalize the well-known niche graphs and are closely related to competition hypergraphs as well as common enemy hypergraphs. For several products D1 ◦ D2 of digraphs D1 and D2, we investigate the relations between the niche hypergraphs of the factors D1, D2 and the niche hypergraph of their product D1 ◦ D2. Keywords: niche hypergraph, product of digraphs, competition hyper- graph. 2010 Mathematics Subject Classification: 05C65, 05C76, 05C20. 1. Introduction and Definitions All hypergraphs H = (V (H), E(H)), graphs G = (V (G),E(G)) and digraphs D =(V (D), A(D)) considered in the following may have isolates but no multiple − + − edges. Moreover, in digraphs loops are forbidden. With ND (v), ND (v), dD(v) + and dD(v) we denote the in-neighborhood, the out-neighborhood, the in-degree 280 M. Sonntag and H.-M. Teichert and the out-degree of v ∈ V (D), respectively. In standard terminology we follow Bang-Jensen and Gutin [1]. In 1968, Cohen [3] introduced the competition graph C(D) = V,E(C(D)) of a digraph D = (V, A) representing a food web of an ecosystem. Here the vertices correspond to the species and different vertices v1, v2 are connected by an edge if and only if they compete for a common prey w, i.e., − − E C(D) = {v1,v2} | v1 = v2 ∧∃ w ∈ V : v1 ∈ ND (w) ∧ v2 ∈ ND (w) . Surveys of the large literature around competition graphs (and its variants) can be found in [5,6,11]; for (a selection of) recent results see [4, 7–10, 12–17, 21]. Meanwhile the following variants of C(D) have been investigated. The common enemy graph CE(D) (cf. [11]) with the edge set + + E CE(D) = {v1,v2} | v1 = v2 ∧∃ w ∈ V : v1 ∈ ND (w) ∧ v2 ∈ ND (w) , the double competition graph or competition-common enemy graph DC(D) with the edge set E(DC(D)) = E(C(D)) ∩ E(CE(D)) (cf. [18]), and the niche graph N(D) with E N(D) = E C(D) ∪ E CE(D) (cf. [2]). In 2004, the concept of competition hypergraphs was introduced by Sonntag and Teichert [19]. The competition hypergraph CH(D) of a digraph D = (V, A) has the vertex set V and the edge set − E CH(D) = e ⊆ V | |e|≥ 2 ∧∃ v ∈ V : e = ND (v) . As a second hypergraph generalization, recently Park and Sano [16] defined the double competition hypergraph DCH(D) of a digraph D =(V, A), which has the vertex set V and the edge set − + E DCH(D) = e ⊆ V | |e|≥ 2 ∧∃ v1,v2 ∈ V : e = ND (v1) ∩ ND (v2) . Our paper [5] was a third step in this direction; there we considered the niche hypergraph NH(D) of a digraph D =(V, A), again with the vertex set V and the edge set − + E NH(D) = e ⊆ V | |e|≥ 2 ∧∃ v ∈ V : e = ND (v) ∨ e = ND (v) . ←− ←− Note that NH(D) = NH D holds for any digraph D, if D denotes the digraph obtained from D by reversing all arcs. In [5] we present results on several properties of niche hypergraphs and the so-called niche number nˆ of hypergraphs. In most of the investigations in [5] the generating digraph D of NH(D) is assumed to be acyclic. Niche Hypergraphs of Products of Digraphs 281 For technical reasons, we define another hypergraph generalization. The common enemy hypergraph CEH(D) of a digraph D =(V, A) has the vertex set V and the edge set + E CEH(D) = e ⊆ V | |e|≥ 2 ∧∃ v ∈ V : e = ND (v) . In the hypergraphs CH(D), CEH(D) and NH(D) no loops are allowed. Therefore, by definition the in-neighborhoods and out-neighborhoods of cardi- nality 1 in the digraph D play no role in the corresponding hypergraphs. This loss of information proved to be disadvantageous in the investigation of com- petition hypergraphs of products of digraphs (cf. [20]). So, considering niche hypergraphs of products of digraphs, it seems to be consequent to allow loops in niche hypergraphs, too. Therefore, we define the l-competition hypergraph CHl(D), the l-common enemy hypergraph CEHl(D) and the l-niche hypergraph NHl(D) (with loops) having the edge sets l − E CH (D) = {e ⊆ V | ∃ v ∈ V : e = ND (v) = ∅}, l + ECEH (D) = e ⊆ V | ∃ v ∈ V : e = ND (v) = ∅ and l − + ENH (D) = e ⊆ V | ∃ v ∈ V : e = ND (v) = ∅∨ e = ND (v) = ∅ l l = E CH (D) ∪ E CEH (D) . For the sake of brevity, in the following we often use the term (l)-competition hypergraph (sometimes in connection with the notation CH(l)(D)) for the compe- tition hypergraph CH(D) as well as for the l-competition hypergraph CHl(D), analogously for (l)-common enemy and (l)-niche hypergraphs with the notations CEH(l)(D) and NH(l)(D), respectively. For five products D1◦D2 (Cartesian product D1×D2, Cartesian sum D1+D2, normal product D1 ∗D2, lexicographic product D1 ·D2 and disjunction D1 ∨D2) of digraphs D1 =(V1, A1) and D2 =(V2, A2) we investigate the construction of the (l) (l) (l) (l) (l)-niche hypergraph NH D1 ◦ D2 = V, E◦ from NH D1 = V1, E1 , (l) (l) NH D2 = V2, E2 and vice versa. The products considered here have always the vertex set V := V1 × V2; using ′ ′ ′ ′ the notation A := {((a, b), (a , b )) | a,a ∈ V1 ∧ b, b ∈ V2} their arc sets are defined as follows: e ′ ′ ′ ′ A D1 × D2 := ((a, b), (a , b )) ∈ A | (a,a ) ∈ A1 ∧ (b, b ) ∈ A2 , ′ ′ ′ ′ ′ ′ AD1+D2 := ((a, b), (a , b )) ∈ A | ((a,a ) ∈ A1∧b = b )∨(a = a ∧(b, b ) ∈ A2) , e AD1 ∗ D2 :=A D1 × D2 ∪ A D1 + D2 , ′ ′ e ′ ′ ′ AD1 · D2 := ((a, b), (a , b )) ∈A | (a,a )∈ A1 ∨ (a = a ∧ (b, b ) ∈ A2) , ′ ′ ′ ′ AD1 ∨ D2 := ((a, b), (a , b )) ∈ A | (a,a ) ∈ A1 ∨ (b, b ) ∈ A2 . e e 282 M. Sonntag and H.-M. Teichert It follows immediately that A D1 + D2 ⊆ A D1 ∗ D2 ⊆ A D1 · D2 ⊆ A D ∨ D and A D × D ⊆ A D ∗ D . Except the lexicographic product 1 2 1 2 1 2 all these products are commutative in the sense that D ◦ D ≃ D ◦ D , where 1 2 2 1 ◦∈{×, +, ∗, ∨}. Usually we number the vertices of D1 and D2 such that V1 = {1, 2,...,r}, V2 = {1, 2,...,s} and arrange the vertices of V = V1 × V2 according to the places of an (r, s)-matrix. In analogy with the rows and the columns of the described (r, s)-matrix we call the set Zi = {(i,j) | j ∈ V2} (i ∈ V1) and the set Sj = {(i,j) | i ∈ V1} (j ∈ V2) the i-th row and the j-th column of D1 ◦ D2, respectively. Then, for each ◦ ∈ {+, ∗, ·, ∨}, the subdigraph SjD1◦D2 of D1 ◦ D2 induced by the vertices of a column Sj is isomorphic to D1, and, analogously, the subdi- graph ZiD1◦D2 of D1 ◦ D2 induced by the vertices of a row Zi is isomorphic to D2. Moreover, if an arc a ∈ A D1 ◦ D2 consists only of vertices of one row Zi (i ∈ V ), we refer to a as a horizontal arc. Analogously, an arc a containing only 1 vertices of one column Sj (j ∈ V2) is called a vertical arc. Considering (l)-niche hypergraphs, the question arises, whether or not (l) (l) (l) NH D1 ◦ D2 can be obtained from NH (D1) and NH (D2) and vice versa. As an instance for competition hypergraphs CH(l), we cite two results from [20]. l l Theorem 1 [20]. The l-competition hypergraph CH (D1 × D2) = V, E× of the l Cartesian product can be obtained from the l-competition hypergraphs CH (D1)= l l l l l l V1, E1 and CH (D2)= V2, E2 of D1 and D2 : E× = e1 × e2 | e1 ∈E1 ∧ e2 ∈E2 . l l Theorem 2 [20]. The l-competition hypergraph CH (D1 ∨ D2) = V, E∨ of l the disjunction can be obtained from the l-competition hypergraphs CH (D1) = l l l V1, E1 and CH (D2)= V2, E2 of D1 and D2, if for each of the following con- ditions is known whether it is true or not: − − (a) ∃ v2 ∈ V2 : N2 (v2)= ∅ and (b) ∃ v1 ∈ V1 : N1 (v1)= ∅. l l l In general, CH (D1 ∨D2) cannot be obtained from CH (D1) and CH (D2) without the extra information on points (a) and (b). Note that in some cases under certain conditions D1 ◦ D2 and even D1 and (l) D2 can be reconstructed from CH (D1 ◦D2). For niche hypergraphs such strong results are not expectable. (l) The main reason why the reconstruction of D1 and D2 from NH (D1 ◦ D2) is much more difficult is the following. In general, for any hyperedge e ∈ (l) − E NH (D) it is not possible to see whether e is a set of predecessors e = ND (v) or a set of successors e = N +(v) of a certain vertex v ∈ V (D). D It is interesting that, in general, for the same reason also the construction of l l NH(D1 ◦ D2) from NH (D1) and NH (D2) is impossible.