Decomposing 1-Sperner hypergraphs

A hypergraph is Sperner if no hyperedge contains another one. A Sperner hypergraph is equilizable (resp., threshold) if the characteristic vectors of its hyperedges are the (minimal) binary solutions to a linear equation (resp., inequality) with positive coefficients. These combinatorial notions have many applications and are motivated by the theory of Boolean functions and integer programming. We introduce in this paper the class of 1-Sperner hypergraphs, defined by the property that for every two hyperedges the smallest of their two set differences is of size one. We characterize this class of Sperner hypergraphs by a decomposition theorem and derive several consequences from it. In particular, we obtain bounds on the size of 1-Sperner hypergraphs and their transversal hypergraphs, show that the characteristic vectors of the hyperedges are linearly independent over the reals, and prove that 1-Sperner hypergraphs are both threshold and equilizable. The study of 1-Sperner hypergraphs is motivated also by their applications in graph theory, which we present in a companion paper.


Introduction
In this paper we consider various classes of hypergraphs, with a focus on the newly introduced class of 1-Sperner hypergraphs.As we will see, this is an interesting and useful notion with many surprising properties, including a simple recursive structure.Before we explain and motivate our study and results, we overview the necessary background definitions.

Background
A hypergraph H is a pair (V, E) where V = V (H) is a finite set of vertices and E = E(H) is a set of subsets of V , called hyperedges [6].Given a positive integer k, a hypergraph H is said to be k-uniform if |e| = k for all e ∈ E(H), and uniform if it is k-uniform for some k.In particular, the (finite, simple, and undirected) graphs are precisely the 2-uniform hypergraphs.Four properties of hypergraphs will be particularly relevant for our study: Sperner, threshold, equilizable, and dually Sperner hypergraphs. 1 Sperner hypergraphs.A hypergraph is said to be Sperner if no hyperedge contains another one, that is, if e, f ∈ E and e ⊆ f implies e = f ; see, e.g., Berge and Duchet [5], Shapiro [40], Sperner [42].Sperner hypergraphs were studied in the literature under different names including simple hypergraphs by Berge [6], clutters by Billera [7,8] and by Edmonds and Fulkerson [19,22], and coalitions in the game theory literature [43].See also [26] for additional references on applications of Sperner hypergraphs in other areas of mathematics.
Threshold hypergraphs.A hypergraph H = (V, E) is said to be threshold if there exist a non-negative integer weight function w : V → Z ≥0 and a non-negative integer threshold t ∈ Z ≥0 such that for every subset X ⊆ V , we have w(X) := x∈X w(x) ≥ t if and only if e ⊆ X for some e ∈ E. A pair (w, t) as above will be referred to as a threshold separator of H.The mapping that takes every hyperedge e ∈ E to its characteristic vector χ e ∈ {0, 1} V , defined by shows that the sets of hyperedges of threshold Sperner hypergraphs are in a one-to-one correspondence with the sets of minimal feasible binary solutions of the linear inequality w x ≥ t.A set of vertices X ⊆ V in a hypergraph is said to be independent if it does not contain any hyperedge, and dependent otherwise.Thus, threshold hypergraphs are exactly the hypergraphs admitting a linear function on the vertices separating the characteristic vectors of the independent sets from the characteristic vectors of dependent sets.Threshold hypergraphs were defined in the uniform case by Golumbic [24] and studied further by Reiterman et al. [38].The 2-uniform threshold hypergraphs are precisely the threshold graphs, introduced by Chvátal and Hammer [16] and studied afterwards in numerous papers; see also the monograph by Mahadev and Peled [28].In their full generality (that is, without the restriction that the hypergraph is uniform), the concept of threshold hypergraphs is equivalent to that of threshold monotone Boolean functions.Threshold Boolean functions provide a simple but fundamental model for many questions investigated in a variety of areas including electrical engineering, artificial intelligence, game theory, cryptography, and many others; see, e.g., Beimel and Weinreb [3], Crama and Hammer [17,18], and Muroga [33].
Close interrelations between hypergraphs and monotone Boolean functions are often useful in the study of threshold and other hypergraphs, allowing for the transfer and applications of results from the theory of Boolean functions; see [18].For example, a polynomial-time recognition algorithm for threshold monotone Boolean functions represented by their complete DNF was given by Peled and Simeone [37].The algorithm is based on linear programming and implies a polynomial-time recognition algorithm for threshold hypergraphs.To the best of our knowledge, no 'purely combinatorial' polynomial-time recognition algorithm for threshold hypergraphs is known [18]. 1   Equilizable hypergraphs.Replacing a linear inequality with positive coefficients by a linear equation maps the notion of threshold hypegraphs to the notion of equilizable hypergraphs.A hypergraph H = (V, E) is said to be equilizable if there exist a (strictly) positive integer weight function w : V → Z >0 and a non-negative integer threshold t ∈ Z ≥0 such that for every subset X ⊆ V , we have w(X) = t if and only if X ∈ E.
Depending on the context, one may want to relax the assumption that all the weights are strictly positive to allow zero weights.However, we find the assumption of strictly positive weights useful for our study; in particular, it implies the following.Proposition 1.1.Every equilizable hypergraph is Sperner.
Proof.Let H = (V, E) be an equilizable hypergraph and let w : V → Z >0 and t ∈ Z ≥0 be such that for every subset X ⊆ V , we have w(X) = t if and only if X ∈ E. Suppose for a contradiction that H is not Sperner.Then, there exist two hyperedges e and f of H such that e ⊂ f .Consequently, w(e) = t = w(f ), which implies that w(f \ e) = 0, contrary to the fact that w is strictly positive.
Equilizable hypergraphs are a very natural family.The sets of hyperedges of an equilizable hypergraph are in a one-to-one correspondence with the sets of binary solutions to a linear equality of the form w x = t where w ∈ Z V >0 and t ∈ Z ≥0 , that is, with the sets of binary vectors that a single hyperplane with positive coefficients can cut out from the hypercube.It is thus not surprising that properties of equilizable hypergraphs are fundamental in integer (or binary) programming.In particular, an old result of Mathews [30] shows how to reduce two linear Diophantine equations with strictly positive coefficients within an arbitrary set to a single equivalent linear equation of the same type.Motivated by integer programming considerations, many authors generalized Mathews' result in a variety of ways, see Anthonisse [2], Bradley [11], Elmaghraby and Wig [21], Glover and Woolsey [23], Padberg [34], and Rosenberg [39].2Furthermore, Mathews' result implies that the class of equilizable hypergraphs on a given vertex set is closed under intersection. 3A related question about linear inequalities led Chvátal and Hammer [16] to the introduction of threshold graphs.
Equilizable hypergraphs can also be seen as a generalization of the class of equistable graphs, defined as follows.A stable set (or: independent set) in a graph G is a set of pairwise non-adjacent vertices.A graph G = (V, E) is said to be equistable if there exist a (strictly) positive integer weight function w : V → Z >0 and a non-negative integer threshold t such that for every subset X ⊆ V , we have w(X) = t if and only if X is an (inclusion-)maximal stable set of G. Equistable graphs were introduced in 1980 by Payan [35], who proved that every threshold graph is equistable.While equistable graphs were originally defined using a function ϕ : V → R ≥0 such that for every subset X ⊆ V , we have ϕ(X) = 1 if and only if X is a maximal stable set of G, it is not difficult to see that the above two definitions are equivalent.Equistable graphs were studied in a series of papers [1,9,25,27,29,31,32,36].However, unlike threshold graphs, the structure of equistable graphs is not understood and the complexity of the problem of recognizing equistable graphs is open.The connection between equistable graphs and equilizable hypergraphs can be easily explained using the notion of stable set hypergraphs.The stable set hypergraph of a graph G is the hypergraph S(G) with vertex set V (G) and in which the hyperedges are exactly the maximal stable sets of G. Clearly, a graph G is equistable if and only if its stable set hypergraph is equilizable.
Dually Sperner hypergraphs.Sperner hypergraphs can be equivalently defined as the hypergraphs such that every two distinct hyperedges e and f satisfy This observation motivated Chiarelli and Milanič to call in [13] a hypergraph H dually Sperner if every two distinct hyperedges e and f satisfy The following result was shown in [13].

The main definition
The main notion studied in this paper is given by the following.
Definition 1.3.Given a positive integer k, we say that a hypergraph H is k-Sperner if every two distinct hyperedges e and f satisfy In particular, H is 1-Sperner if every two distinct hyperedges e and f satisfy or, equivalently, if, for any two distinct hyperedges e and f of Denoting by S k the class of all k-Sperner hypergraphs and by S the class of all Sperner hypergraphs, it is clear that these families of hypergraphs are related by the following chain of inclusions The inclusions follow immediately from the definitions, while the equality follows from (1).Moreover, since we do not allow multiple hyperedges, every 2-uniform hypergraph (that is, a graph) is k-Sperner for every k ≥ 2. We should therefore not expect useful decomposition properties for the classes of k-Sperner hypergraphs for k ≥ 2. We focus in this paper on the case k = 1 and show that hypergraphs in the corresponding subfamily S 1 have a nice structure.Note that by definition, all hypergraphs with at most one hyperedge (possibly with no vertices) are 1-Sperner.Note also that a hypergraph is 1-Sperner if and only if it is both Sperner and dually Sperner.
The concept of 1-Sperner hypergraphs already appeared in some graph theoretical research.Chiarelli and Milanič [12,13] made use of dually Sperner hypergraphs to characterize two classes of graphs defined by the following properties: every induced subgraph has a non-negative linear vertex weight function separating the characteristic vectors of all total dominating sets [12], resp.connected dominating sets [13], from the characteristic vectors of all other sets.Due to the close relation between 1-Sperner and dually Sperner hypergraphs (see Observation 2.9), all the results from [12,13] can be equivalently stated using 1-Sperner hypergraphs.In particular, the results of the extended abstract [12] are stated using the 1-Sperner property in the full version of the paper [14].

Our results
Our main result is a decomposition theorem for 1-Sperner hypergraphs.We also derive several consequences of it.
The decomposition theorem.We define a simple operation on hypergraphs called gluing and show that it produces (with only one small exception) a new 1-Sperner hypergraph from a given pair of 1-Sperner hypergraphs; see Fig. 2 for an example illustrated with incidence matrices.Conversely, we show that every 1-Sperner hypergraph with at least one vertex is the gluing of two smaller 1-Sperner hypergraphs; see Theorem 4.2.
Consequences.We use the decomposition theorem to prove the following properties of 1-Sperner hypergraphs.a) Every 1-Sperner hypergraph is threshold and has a positive threshold separator; see Theorem 5.2.
In particular, this gives a new, constructive proof of the fact that every dually Sperner hypergraph is threshold, obtained first by Chiarelli and Milanič [13]; see Theorem 1.2.
c) The characteristic vectors of the hyperedges of a 1-Sperner hypergraph are linearly independent over the reals; see Theorem 5.6.This implies that the number of hyperedges cannot exceed the number of vertices, thus giving a sharp upper bound on the size of a 1-Sperner hypergraph in terms of its order; see Corollary 5.7.We also give a sharp lower bound on the size of a 1-Sperner hypergraph without universal, isolated, and twin vertices, in terms of its order; see Proposition 5.8.
d) The number of minimal transversals of a 1-Sperner hypergraph is bounded from above by a quadratic function of its order and they can be efficiently generated; see Theorem 5.9.
Our study of 1-Sperner hypergraphs is motivated not only by their nice combinatorial properties but also by their numerous applications in graph theory.Some of them were already mentioned above and we obtained several others.To keep the length of this paper reasonable, we decided to present those results in a separate paper [10].We briefly summarize them here.
We use the characterizations of so-called threshold and domishold graphs in terms of forbidden induced subgraphs due to Chvátal and Hammer [16] and Benzaken and Hammer [4], respectively, to derive further characterizations of these graph classes in terms of 1-Spernerness, thresholdness, and 2-asummability properties of several related hypergraphs, namely their vertex cover, clique, independent set, dominating set, and closed neighborhood hypergraphs.
Furthermore, we use the decomposition theorem for 1-Sperner hypergraphs (Theorem 4.2) to derive decomposition theorems for four classes of graphs, namely two classes of split graphs, a class of bipartite graphs, and a class of cobipartite graphs.These decomposition theorems are based on certain matrix partitions of the corresponding graphs and give rise to new classes of graphs of bounded clique-width and to new polynomially solvable cases of variants of domination.

Interrelations between the considered classes of hypergraphs
In Fig. 1, we show the Hasse diagram of the partial order of the hypergraph classes studied in this paper, ordered with respect to inclusion.
The fact that every 1-Sperner hypergraph is threshold and equilizable is proved in Theorems 5.2 and 5.3, respectively.The fact that every dually Sperner hypergraph is threshold was proved by Chiarelli and Milanič [13].The fact that every threshold graph is equistable was proved by Payan [35].The fact that every equilizable hypergraph is Sperner was proved in Proposition 1.1.The remaining inclusions are trivial.
Finally, the following examples show that all inclusions are strict and there are no other inclusions: • the complete graph K 4 is a 2-uniform hypergraph that is threshold but not dually Sperner; • the hypergraph with vertex set {1, 2, 3} and hyperedge set {{1, 2, 3}} is 1-Sperner but not 2-uniform; • the hypergraph with vertex set {1} and hyperedge set {∅, {1}} is dually Sperner but not Sperner, • an equilizable hypergraph that is not threshold is presented in Example 5.1; • a 2-uniform threshold hypergraph that is not equilizable is presented in Example 5.2; • a threshold and equilizable hypergraph that is neither dually Sperner nor 2-Sperner is the complete 3-uniform hypergraph H 6,3 ; see Example 5.3; • the cycle C 4 is an equistable graph that, when viewed as a 2-uniform hypergraph, it is not threshold [16]; • the path P 4 is a graph that is not equistable [35]; moreover, it is also a Sperner hypergraph that is not equilizable.
The remaining non-inclusions follow by transitivity.
Structure of the paper.In Section 2, we collect the necessary definitions and preliminary results.
We also consider several operations on hypergraphs and show that the class of 1-Sperner hypergraphs is (almost always) closed under these operations.In Section 3 we give a necessary condition for a uniform hypergraph to be 1-Sperner and identify two families of uniform 1-Sperner hypergraphs.
Building on the results of Sections 2 and 3, we develop in Section 4 the composition theorem for 1-Sperner hypergraphs.Various consequences of this theorem are examined in Section 5.

Definitions and hypergraph operations
The order and the size of a hypergraph H refer to the number of its vertices, resp.hyperedges.Every hypergraph H = (V, E) with a fixed pair of orderings of its vertices and edges, say V = {v 1 , . . ., v n }, and E = {e 1 , . . ., e m }, can be represented with its incidence matrix A H ∈ {0, 1} E×V having rows and columns indexed by edges and vertices of H, respectively, and defined as Note the slight abuse of notation above: the incidence matrix does not depend only on the hypergraph but also on the pair of orderings of its vertices and edges.We will be able to neglect this technical issue often in the paper, but not always.We will therefore say that two matrices A and B of the same dimensions are permutation equivalent, and denote this fact by A ∼ = B, if A can be obtained from B by permuting some of its rows and/or columns.For later use, we state a simple property of incidence matrices of a hypergraph.
Remark 1.Let H = (V, E) be a hypergraph with a fixed pair of orderings of its vertices and edges, respectively, and let A H be the corresponding incidence matrix.Then, any permutation of the vertices and/or edges of H results in an incidence matrix that is permutation equivalent to A H .Moreover, any matrix that is permutation equivalent to A H is the incidence matrix of H with respect to some pair of orderings of its vertices and edges.
k-asummable hypergraphs.A hypergraph is k-asummable if it has no k (not necessarily distinct) independent sets A 1 , . . ., A k and k (not necessarily distinct) dependent sets B 1 , . . ., B k such that The following characterization of threshold graphs follows from analogous characterizations of threshold monotone Boolean functions; see [18].
Next we consider several operations on hypergraphs and show that the class of 1-Sperner hypergraphs is (almost always) closed under these operations.
Proof.This follows directly from the definition, using the fact that for every two sets e, f ⊆ V , we have e \ f = f \ e and f \ e = e \ f .As the next example shows, the closure under complementation does not hold for the classes of threshold Sperner hypergraphs and 2-asummable Sperner hypergraphs.

Gluing of hypergraphs
The decomposition theorem (Theorem 4.2) is based on the following general operation.

Definition 2.3 (Gluing of two hypergraphs). Given a pair of vertex-disjoint hypergraphs
Let us note the operation of gluing is well-defined also if some of the sets V 1 , V 2 , E 1 , and E 2 are empty.The operation can be visualized easily in terms of incidence matrices.Let n i = |V i | and m i = |E i | for i = 1, 2, and let us denote by 0 k, , resp. 1 k, , the k × matrix of all zeroes, resp. of all ones.Then, the incidence matrix of the gluing of H 1 and H 2 can be written as See Fig. 2 for an example.To further illustrate the operation of gluing, let us note that the operation generalizes the operations of adding an isolated or a universal vertex.A vertex u in a hypergraph H = (V, E) is said to be universal (resp., isolated ) if it is contained in all (resp., in no) hyperedges.The operations of adding an isolated or a universal vertex to a hypergraph are defined in the natural way.
Observation 2.4.For every hypergraph H, the following holds: • The hypergraph obtained by adding an isolated vertex to H is the result of gluing of (∅, ∅) and H.
• The hypergraph obtained by adding a universal vertex to H is the result of gluing of H and (∅, ∅).
The gluing and the complementation operations are related as follows (the proof is left as an easy exercise for the reader): assuming that in both gluing operations the same new vertex is used).
It is easy to see that if the gluing of H 1 and H 2 is a 1-Sperner hypergraph, then both constituent hypergraphs are 1-Sperner.We record this fact for later use.
Observation 2.6.If the gluing of H 1 and H 2 is a 1-Sperner hypergraph, then H 1 and H 2 are also 1-Sperner.
The next proposition establishes a partial converse.Gluing preserves 1-Spernerness, unless the resulting hypergraph is not Sperner, which happens only in one very special case.Proposition 2.7.For every pair Proof.Let e and f be two distinct edges of H 1 H 2 .If z ∈ e ∩ f then their differences are the same as the corresponding differences of e \ {z} and f \ {z}, both of which are hyperedges of H 1 .If z ∈ e ∪ f then their differences are the same as the corresponding differences of e \ V 1 and f \ V 1 , both of which are hyperedges of H 2 .If z ∈ e \ f , then e \ f = {z} and f \ e = ∅, unless e = V 1 , and f = ∅ (which implies E 1 = {V 1 } and E 2 = {∅} by our assumption that both H 1 and H 2 are 1-Sperner).The case of z ∈ f \ e is symmetric.

Sperner reductions and hypergraph transversals
Given a hypergraph H = (V, E), its Sperner reduction, Sp(H), is the hypergraph with vertex set V and with hyperedges the inclusion-minimal elements of E.
The following observation is easy to prove from the definitions.
Observation 2.8.For any two hypergraphs The following observation is a direct consequence of the definitions of dually Sperner and 1-Sperner hypergraphs.
Furthermore, the problem of studying the thresholdness property in a class of hypergraphs reduces to the class of their Sperner reductions.
Proposition 2.10.Let H = (V, E) be a hypergraph, let w : V → Z ≥0 and t ∈ Z ≥0 .Then, (w, t) is a threshold separator of H if and only if (w, t) is a threshold separator of Sp(H).In particular, H is threshold if and only if its Sperner reduction is threshold.
Proof.Let us call a subset of vertices X ⊆ V heavy if w(X) ≥ t, and light, otherwise.The pair (w, t) is a threshold separator of H if and only if the heavy subsets of V are precisely those containing a hyperedge of H. Since the set of heavy subsets depends only on (w, t) and not on H and a subset of V contains a hyperedge of H if and only if it contains a hyperedge of Sp(H), the proposition follows.
Let H = (V, E) be a hypergraph.A transversal of H is a set of vertices intersecting all hyperedges of H.The transversal hypergraph H T is the hypergraph with vertex set V in which a set X ⊆ V is a hyperedge if and only if X is an inclusion-minimal transversal of H. (In particular, if H has no hyperedge, then its transversal hypergraph is H T = (V (H), {∅}).)Observation 2.11 (see, e.g., Berge [6]).If H is a Sperner hypergraph, then (H T ) T = H.
A pair of a mutually transversal Sperner hypergraphs naturally corresponds to a pair of dual monotone Boolean functions, see [18].
The next proposition, which will be used in the proof of Theorem 5.9, describes how to compute the transversal hypergraph of the gluing of two hypergraphs H 1 and H 2 from their transversal hypergraphs.Proposition 2.12.Let H be a gluing of two vertex-disjoint hypergraphs We will first show that every set in F is a transversal of H and then we will argue that every minimal transversal of H appears in F .Together, by Observation 2.8, these two claims will imply the stated equality.
The first claim is easy to see by the definition of the gluing operation.
For the second claim, let X be a minimal transversal of H. Suppose first that E 1 = ∅.Note that in this case z is an isolated vertex of H, so no minimal transversal of H can contain z.
Finally, assume that E 1 = ∅.Suppose also that V 1 = ∅.Then all minimal transversals of H must contain z and must intersect all hyperedges of H 2 .Thus, X must have the form X = {z}cupe for some e ∈ E H T 2 ; in particular In either case, X belongs to F .This completes the proof.

Ungluing hypergraphs
We next introduce some terminology related to hypergraphs that are the result of a gluing operation.Given a vertex z of a hypergraph H, we say that a hypergraph H is z-decomposable if for every two hyperedges e, f ∈ E(H) such that z ∈ e \ f , we have e \ {z} ⊆ f .Equivalently, if the vertex set of H can be partitioned as The following proposition gathers some basic properties of decomposability.
Proposition 2.13.Let H be a hypergraph.Then, the following holds: (ii) If z is an isolated or a universal vertex of H, then H is z-decomposable.
Proof.Statement (i) follows from Observation 2.5.Statement (ii) is related to Observation 2.4.Note that z is universal in H if and only if it is isolated in H.By (i), it therefore suffices to prove the statement for the case when z is an isolated vertex of H.In this case, the column of A H indexed by z is the all zero vector.It follows that H is z-decomposable, as follows: Recall that by Observation 2.6, if a 1-Sperner hypergraph H has a z-decomposition H = H 1 H 2 , then H 1 and H 2 are also 1-Sperner.
Whether a given hypergraph is z-decomposable for some vertex z can be checked in a straightforward way in polynomial time.
Proposition 2.14.Let H = (V, E) be a hypergraph with V = ∅ and E = ∅ given by the lists of its vertices and hyperedges.We can recognize if H is z-decomposable for some z ∈ V and find a corresponding z-decomposition Proof.It suffices to show that for a given vertex z ∈ V we can verify in time O(|V ||E|) if H is z-decomposable and find a corresponding z-decomposition H = H 1 H 2 (if there is one).
First, we partition the hyperedges of H into those containing z and those not containing z.Secondly, we compute the sets E 1 = {e \ {z} | z ∈ e ∈ E} and V 1 = ∪{e | e ∈ E 1 }.Thirdly, we verify if for every hyperedge e ∈ E not containing z, we have V 1 ⊆ e.If this condition is not satisfied, then H is not z-decomposable.If the condition is satisfied, then we compute the sets Clearly, if H = (V, E) is a hypergraph with at least one vertex and no hyperedges, then H is z-decomposable for every z ∈ V and a z-decomposition of H can be computed in time O(|V |).

Uniform 1-Sperner hypergraphs
In the next lemma we give a necessary condition for a uniform hypergraph to be 1-Sperner.The condition will be used in the proof of Theorem 4.2.Lemma 3.1.Let H be a k-uniform 1-Sperner hypergraph, where k ≥ 1.Then, either there is a subset P of vertices of size k − 1 such that P ⊆ e for all e ∈ E(H) or there is a subset Q of vertices of size k + 1 such that e ⊆ Q for all e ∈ E(H).
Proof.The statement of the lemma holds if H has at most one hyperedge.So let us assume that H has at least two hyperedges, say e and f .Let P = e ∩ f .Since H is 1-Sperner, |P | = k − 1.If all hyperedges of H contain P , then we are done.
If there is a hyperedge g such that P g, say u ∈ P \ g, then e and f are the only hyperedges containing P , since otherwise g should contain all vertices of such hyperedges other than u, which would imply |g| > k.Consequently, all hyperedges that miss a vertex of P are subsets of Q = e ∪ f , and the lemma is proved.Lemma 3.1 suggests the following two families of uniform 1-Sperner hypergraphs.
and X ∩ Y = ∅, and If this is the case, we say that H is the (k-)star generated by (V, X, Y ).
Clearly, every k-star is 1-Sperner.Moreover, let us verify that every k-star is z-decomposable with respect to every vertex z.Let H be a k-star generated by (V, X, Y ) and let z ∈ V (H).If z ∈ X, then k ≥ 2 and we have H = H 1 H 2 where H 1 is the (k − 1)-star generated by X \ {z} and Y and V (H 2 ) = E(H 2 ) = ∅.If z ∈ Y , then we have H = H 1 H 2 where H 1 = (X, {X}) and If this is the case, we say that H is the (k-)antistar generated by (V, X, Y ).Note that every k-antistar is the complement of a k-star.It follows, using Propositions 2.2 and 2.13 and the properties of stars observed in Example 3.1, that every antistar is 1-Sperner and z-decomposable with respect to each vertex z.

Decomposition theorem
To prove the main structural result about 1-Sperner hypergraph (Theorem 4.2), we need the following technical lemma.Proof.Note that |A| ≤ |C|, therefore A \ C = {x}, since H is 1-Sperner.Analogously, B \ C = {y}.Thus, if the sets A ∩ C and B ∩ C were not comparable with respect to inclusion, the pair {A, B} would violate the 1-Sperner property of H. Theorem 4.2.Every 1-Sperner hypergraph H = (V, E) with V = ∅ is z-decomposable for some z ∈ V (H), that is, it is the gluing of two 1-Sperner hypergraphs.
Proof.By Proposition 2.13, we may assume that H does not have any isolated vertices.For every We consider two cases.
Case 1: Not all the k(v) values are the same.Let v ∈ V be a vertex with the smallest k(v) value.
Then k(v) < k by the assumption of this case.Suppose first that for every hyperedge f ∈ E such that v ∈ f , we have |f | ≥ k(v).We claim that in this case H is v-decomposable.This is because for every two hyperedges e, f from what we derive, using the fact that H is 1-Sperner, that |e \ f | = 1, that is, e \ {v} ⊆ f .This proves the claim.
Assume next that there exists a hyperedge f ∈ E such that v ∈ f and |f | < k(v).Let e be a hyperedge containing v of size k(v), and let g be a hyperedge of maximum size, that is, We know that k(u) ≥ k(v), by our choice of v. Therefore, there exists a hyperedge h containing u and of size k(u).
It follows that f ⊆ h, contradicting the Sperner property of H.This completes Case 1.
Case 2: All the k(v) values are the same.Let v ∈ V (H) and let k = k(v).If k ≤ 1, then H is z-decomposable with respect to every vertex z.So suppose that k ≥ 2. Consider the subhypergraph H of H with V (H ) = V (H) formed by the hyperedges of H of size k.By Lemma 3.1 applied to H , either there is a subset P of vertices of size k − 1 such that P ⊆ e for all e ∈ E(H ) or there is a subset Q of vertices of size k + 1 such that e ⊆ Q for all e ∈ E(H ).
Suppose first that there is a subset P of vertices of size k − 1 such that P ⊆ e for all e ∈ E(H ).If H = H, that is, all hyperedges of H are of size k, then H is z-decomposable with respect to every vertex z (cf.Example 3.1).So we may assume that H = H, that is, that H contains a hyperedge g of size less than k.By the assumption of Case 2, we know that g ⊆ ∪ f ∈E(H ) f .Since H is Sperner, g is not contained in any of the hyperedges of H ; moreover g contains at least two vertices from the set Y = ∪ f ∈E(H ) f \ P .If g contains at least three vertices from Y , say y 1 , y 2 , y 3 , then the hyperedges P ∪ {y 1 } and g would violate the 1-Sperner property, since {y 2 , y 3 } ⊆ g \ (P ∪ {y 1 }) and In fact, we have |Y | = 2, say Y = {y 1 , y 2 }, since otherwise, using similar arguments as above, we see that the sets P ∪ {y} and g would violate the 1-Sperner property, where y ∈ Y \ g.It follows that H has exactly 2 hyperedges, and Y ⊆ e for every set e ∈ E(H) \ E(H ).Consequently, H is y-decomposable for every y ∈ Y : Decomposing H with respect to y = y 1 , for instance, we have It remains to consider the case when there is a subset Q of vertices of size k + 1 such that e ⊆ Q for all e ∈ E(H ).Since we assume that k(v) = k for all vertices v, we have Then, Y = ∅, and every hyperedge g ∈ E(H) \ E(H ) must contain Y , since H is Sperner and for every vertex y ∈ Y , the set V \ {y} is a hyperedge of H. Consequently, H is y-decomposable for every y ∈ Y : taking any y ∈ Y , we have Let us say that a gluing of two vertex-disjoint 1-Sperner hypergraphs is safe if it results in a 1-Sperner hypergraph.By Proposition 2.7, this is always the case unless E 1 = {V 1 } and E 2 = {∅}.Thus, Theorem 4.2 and Proposition 2.7 imply the following composition result for the class of 1-Sperner hypergraphs.

Applications of the decomposition theorem
In this section we present several applications of Theorems 4.2 and 4.3 giving further insight on the properties of 1-Sperner hypergraphs.

1-Sperner hypergraphs are threshold
Our first application is motivated by the result of Chiarelli and Milanič [13] stating that every dually Sperner hypergraph is threshold, see Theorem 1.2.The proof of Theorem 1.2 given in [13] is based on the characterization of thresholdness in terms of asummability (see Theorem 2.1) and does not show how to compute a threshold separator of a dually Sperner hypergraph.Here we give an alternative proof of Theorem 1.2, based on the composition theorem of 1-Sperner hypergraphs.In contrast with the proof from [13], this proof is constructive in the sense that it computes an explicit threshold separator of a 1-Sperner or, more generally, dually Sperner hypergraph.
Clearly, every 1-Sperner hypergraph is dually Sperner.Therefore, Theorem 1.2 implies that every 1-Sperner hypergraph is threshold.We will now derive this fact directly from the composition theorem.In fact, we show a bit more, namely that every 1-Sperner hypergraph H = (V, E) admits a positive threshold separator, that is, a threshold separator (w, t) such that w : V → Z >0 is a (strictly) positive integer weight function.
The following simple technical claim will be used in the proof.
Lemma 5.1.For every threshold separator (w, t) of a Sperner threshold hypergraph H = (V, E), we have: Proof.If w(V ) = t, then V is a hyperedge and if t = 0, then the empty set is a hyperedge.(Both of these claims follow from the fact that (w, t) is a threshold separator of H.) In both cases no other hyperedge may exist due to the Sperner property.
Proof.Let H = (V, E) be a 1-Sperner hypergraph.The proof is by induction on n = |V |.For n = 0, we can obtain a positive threshold separator by taking the (empty) mapping given by w(x) = 1 for all x ∈ V and the threshold Now, let n ≥ 1.By Theorem 4.3, H is the safe gluing of two 1-Sperner hypergraphs, say By the inductive hypothesis, H 1 and H 2 admit positive threshold separators.That is, there exist positive integer weight functions w i : V i → Z >0 and non-negative integer thresholds t i ∈ Z ≥0 for i = 1, 2 such that for every subset X ⊆ V i , we have w i (X) ≥ t i if and only if e ⊆ X for some e ∈ E i .
Let us define the threshold t = M w 1 (V 1 ) + t 2 , where M = w 2 (V 2 ) + 1, and the weight function w : V → Z >0 by the rule We claim that (w, t) is a positive threshold separator of H. Let us first verify that the so defined weight function is indeed positive.Since w i for i ∈ {1, 2} are positive and M > 0, we have w(x) > 0 for all x ∈ V 1 ∪ V 2 .Moreover, since w 1 (V 1 ) ≥ t 1 , M ≥ 0, and t 2 ≥ 0, we have w(z) ≥ 0. If w(z) = 0, then w 1 (V 1 ) = t 1 and t 2 = 0, which by Lemma 5.1 implies E 1 = {V 1 } and E 2 = {∅}, contrary to the fact that the gluing is safe.It follows that w(z) > 0, as claimed.
Next, we verify that (w, t) is a threshold separator of H, that is, that for every subset X ⊆ V , we have w(X) ≥ t if and only if e ⊆ X for some e ∈ E.
Suppose first that w(X) ≥ t for some X ⊆ V .Let X i = X ∩ V i for i = 1, 2. For later use, we note that Suppose first that z ∈ X.Then If w 1 (X 1 ) ≤ t 1 − 1 then, using (2), we obtain a contradiction with (3).It follows that w 1 (X 1 ) ≥ t 1 .Consequently there exists e 1 ∈ E 1 such that e 1 ⊆ X 1 , hence the hyperedge e := {z} ∪ e 1 ∈ E satisfies e ⊆ X.Now, suppose that z ∈ X.In this case, We must have where the last inequality follows from (2) and t 2 ≥ 0. Therefore, inequality (7) simplifies to w 2 (X 2 ) ≥ t 2 , and consequently there exists a hyperedge e 2 ∈ E 2 such that e 2 ⊆ X 2 .This implies that H has a hyperedge e For the converse direction, suppose that X is a subset of V such that e ⊆ X for some e ∈ E. We need to show that w(X) ≥ t.We consider two cases depending on whether z ∈ e or not.Suppose first that z ∈ e.Then e = {z} ∪ e 1 for some e 1 ∈ E 1 .Due to the property of w 1 , we have w 1 (e 1 ) ≥ t 1 .Consequently, Suppose now that z ∈ e.Then e = V 1 ∪ e 2 for some e 2 ∈ E 2 .Due to the property of w 2 , we have w 2 (e 2 ) ≥ t 2 .Consequently, This shows that w(X) ≥ t whenever X contains a hyperedge of H, and completes the proof.
We now give an alternative proof of Theorem 1.2 announced above.An alternative proof of Theorem 1.2.Let H = (V, E) be a dually Sperner hypergraph.By Observation 2.9, its Sperner reduction is 1-Sperner.By Theorem 5.2, Sp(H) has a positive threshold separator, say (w, t).Since (w, t) is a threshold separator of Sp(H), it is also a threshold separator of H, by Proposition 2.10.Thus, H is threshold.
We would like to emphasize that the above proof implies the following simple efficient procedure of obtaining a threshold separator of a given dually Sperner hypergraph H: (1) compute its Sperner reduction, Sp(H), and (2) construct a positive threshold separator (w, t) of Sp(H) recursively along a decomposition of Sp(H) into smaller 1-Sperner hypergraphs given by Theorem 4.3 (eventually resulting in trivial 1-Sperner hypergraphs).
Then (w, t) is a threshold separator of H.

Further relations between threshold, equilizable, and 1-Sperner hypergraphs
The same inductive construction of a threshold separator as that given in the proof of Theorem 5.2 shows that every 1-Sperner hypergraph is also equilizable (see Section 1.1 for the definition).
Theorem 5.3 can be proved by slightly modifying the above proof of Theorem 5.2; for the sake of completeness, we include it in Appendix.Theorem 5.3 will be used in Section 5.3 to establish an upper bound on the size of a 1-Sperner hypergraph of a given order.
Combining Theorems 5.2 and 5.3 shows that every 1-Sperner hypergraph is threshold and equilizable.In particular, the properties of thresholdness and equilizability trivially coincide within the class of 1-Sperner hypergraphs.This raises the question of whether the two properties are comparable in the larger class of Sperner hypergraphs.This is not the case.As the following two examples show, the properties of thresholdness and equilizability are incomparable in the class of Sperner hypergraphs.
Example 5.1.The following Sperner hypergraph is equilizable but not threshold: defined by w(v 1 ) = 5, w(v 2 ) = 4, w(v 3 ) = 3, w(v 4 ) = 2, and w(v 5 ) = 7 assigns a total weight of 9 to each hyperedge and to no other subset of V 1 .Thus, H 1 is equilizable.To see that H 1 is not threshold, note that any threshold separator (w , t ) of H 1 would have to satisfy w (v 1 ) + w (v 2 ) ≥ t and w (v 3 ) + w (v 4 ) ≥ t , as well as w (v 1 ) + w (v 3 ) < t and w (v 2 ) + w (v 4 ) < t , which is impossible.In other words, H 1 fails to be threshold since it is not 2-asummable; cf.Theorem 2.1.
Example 5.2.The following Sperner hypergraph is threshold but not equilizable: , and threshold t = 3 form a threshold separator of H 2 .Thus, H 2 is threshold.To see that H 2 is not equilizable, note that any function w : V 2 → Z ≥0 such that the total weight of every hyperedge is the same, say t , must assign weight t /2 to every vertex.Consequently, the set {v 1 , v 4 }, which is not a hyperedge, would also be of total weight t .
Furthermore, the following examples show that there exist Sperner hypergraphs that are threshold and equilizable but not 1-Sperner.

Bounds on the size of 1-Sperner hypergraphs
We now establish some upper and lower bounds on the number of hyperedges in a 1-Sperner hypergraph with a given number of vertices.By 0, resp. 1, we will denote the vector of all zeroes, resp.ones, of appropriate dimension (which will be clear from the context).The following lemma can be easily derived from Theorem 5.3.Lemma 5.4.For every 1-Sperner hypergraph H = (V, E) such that E = ∅ and E = {∅}, there exists a vector x ∈ R V ≥0 such that A H x = 1 and 1 x ≥ 1.
Proof.Let H = (V, E) be a 1-Sperner hypergraph as in the statement of the lemma.By Theorem 5.3, H is equilizable.Let w : V → Z ≥0 be a non-negative integer weight function and t ∈ Z ≥0 a non-negative integer threshold such that for every subset X ⊆ V , we have w(X) = t if and only if X ∈ E. If t = 0, then ∅ ∈ E and consequently E = {∅}, a contradiction.It follows that t > 0, and we can define the vector x ∈ R V ≥0 given by x v = w(v)/t for all v ∈ V .We claim that vector x satisfies the desired properties A H x = 1 and 1 x ≥ 1.
Since w(X) = t for all X ∈ E, we have A H x = 1.Since E = ∅, an arbitrary hyperedge e ∈ E shows that t = w(e) ≤ w(V ).Consequently, we also have Corollary 5.5.For every 1-Sperner hypergraph H = (V, E) and every vector λ ∈ R E we have Proof.If E = ∅ or E = {∅}, then the left hand side of the above implication is always false.In all other cases, by Lemma 5.4, there exists a vector x ∈ R V such that A H x = 1 and 1 The composition theorem and the above corollary imply the following useful property of 1-Sperner hypergraphs.
Theorem 5.6.For every 1-Sperner hypergraph H = (V, E) such that E = {∅}, the characteristic vectors of its hyperedges are linearly independent (over the field of real numbers). Proof.
Since the characteristic vectors of the hyperedges of an n-vertex 1-Sperner hypergraph are linearly independent vectors in R n , we obtain the following upper bound on the size of a 1-Sperner hypergraph in terms of its order.The bound |E| ≤ |V | can also be proved more directly from the decomposition theorem (Theorem 4.3), using induction on the number of vertices and analyzing various cases according to whether the two constituent hypergraphs have non-empty vertex set or not.We decided to include the proof based on Theorem 5.6, since linear independence is an interesting property of 1-Sperner hypergraphs and the inequality |E| ≤ |V | is just one consequence of that.
We now turn to the lower bound.Recall that a vertex u in a hypergraph H = (V, E) is said to be universal (resp., isolated ) if it is contained in all (resp., in no) hyperedges.Moreover, two vertices u, v of a hypergraph H = (V, E) are twins if they are contained in exactly the same hyperedges.
Corollary 5.7 gives an upper bound on the size of a 1-Sperner hypergraph in terms of its order.Can we prove a lower bound of a similar form?In general not, since adding universal vertices, isolated vertices, or twin vertices preserves the 1-Sperner property and the size, while it increases the order.However, as we show next, for 1-Sperner hypergraphs without universal, isolated, and twin vertices, the following sharp lower bound on the size in terms of the order holds.Proof.We use induction on n = |V |.For n ∈ {2, 3, 4}, it can be easily verified that the statement holds.Now, let H = (V, E) be a 1-Sperner hypergraph with n ≥ 5 and without universal vertices, isolated vertices, and twin vertices.By Theorem 4.2, H is the gluing of two 1-Sperner hypergraphs, say H = H 1 H 2 with H 1 = (V 1 , E 1 ) and H 2 = (V 2 , E 2 ), where We have m = m 1 + m 2 , and by the rules of the gluing, n = n 1 + n 2 + 1.By Proposition 2.2, we may assume that n 1 ≥ n 2 (otherwise, we can consider the complementary hypergraph).In particular, n 1 ≥ 3. The fact that H does not have a universal vertex implies H 1 does not have a universal vertex.Similarly, H 2 does not have an isolated vertex.Since H does not have any pairs of twin vertices, we have that either H 1 does not have a isolated vertex, or H 2 does not have a universal vertex.We may assume that H 2 does not have a universal vertex (otherwise, we consider a different gluing in which we delete the universal vertex from H 2 and add an isolated vertex to H 1 ).Since H 2 is a Sperner hypergraph without an isolated or a universal vertex, we have n 2 = 1.
Suppose first that n 2 ≥ 2. We apply the inductive hypothesis for H 1 and H 2 , where H 1 is the hypergraph obtained from H 1 by deleting from it the isolated vertex (if it exists).Letting n 1 = |V (H 1 )| and m 1 = |E(H 1 )|, we thus have n 1 ≥ n 1 − 1 and also n 1 ≥ 2. We obtain which implies This completes the proof of the inequality.
• For k > 2, we set H k = H k−1 H k−1 where H k−1 is the hypergraph obtained from a disjoint copy of H k−1 by adding to it an isolated vertex.
An inductive argument shows that for every k ≥ 2, we have , and consequently m k = n k 2 + 1 .

Minimal transversals of 1-Sperner hypergraphs
Recall that a transversal of H is a set of vertices intersecting all hyperedges of H.

Figure 2 :
Figure 2: An example of gluing of two hypergraphs.
) and H 2 = (V 2 , E 2 ).Since each of the steps can be performed in time O( e∈E |e|) = O(|V ||E|), the claimed time complexity follows.

Lemma 4 . 1 .
Let H be a 1-Sperner hypergraph with E(H) = ∅ and let C be a hyperedge of H of maximum size.Then, for every two distinct vertices x, y ∈ C and every two hyperedges A containing x and B containing y, |A| ≤ |B| implies A ∩ C ⊆ B ∩ C.
v∈e∈E |e| and let k = max e∈E |e|.

Proposition 5 . 8 .
For every 1-Sperner hypergraph H = (V, E) with |V | ≥ 2 and without universal, isolated, and twin vertices, we have the following sharp lower bound Since H has no twins, H 1 and H 2 also have no twins.Let n i = |V i | and m i = |E i | for i = 1, 2, and let m = |E|.
We use induction on |V |.If |V | ≤ 1, then the statement holds since E = {∅}.Suppose now that |V | > 1.Then by Theorem 4.2, H is the gluing of two 1-Sperner hypergraphs, say H = H 1 H 2 with H 1 Suppose now that n 2 = 0.In this case, since H does not have a universal vertex, we must have E 2 = {∅} and m 2 = 1.As above, let H 1 be the hypergraph obtained from H 1 by deleting from it the isolated vertex (if it exists).Letting n 1 = |V (H 1 )| and m 1 = |E(H 1 )|, we obtain, by applying the inductive hypothesis to H 1 ,