1 Introduction
Hypergraphs are one of the most fundamental and general combinatorial objects, encompassing various important structures such as graphs, matroids, and combinatorial designs. Results for more specific discrete structures (e.g., graphs) can often be proved more generally, in the context of suitable classes of hypergraphs (see, e.g., Schrijver [57]). Of course, applications of hypergraph theory to structural and optimization aspects of other discrete structures are not restricted to the above phenomenon. Since it is impossible to survey here all kinds of applications of hypergraphs, let us mention only a few more. First, recent work of Hujdurović et al. [37]
made use of connections between hypergraphs and binary matrices, acyclic digraphs, and partially ordered sets to study two combinatorial optimization problems motivated by computational biology. Second, a common approach of applying hypergraph theory to graphs is to define and study a hypergraph derived in an appropriate way from a given graph, depending on what type of property of a graph or its vertex or edge subsets one is interested in. This includes matching hypergraphs
[1, 63], various clique [32, 63, 47, 51, 41], independent set [32, 63], neighborhood [27, 12], separator [60, 13], and dominating set hypergraphs [12, 13], etc. Close interrelations between hypergraphs and monotone Boolean functions can be useful in such studies, allowing for the transfer and applications of results from the theory of Boolean functions (see, e.g., [21]).In this work, we present several new applications of hypergraphs to graphs. Our starting point is [9] where the class of Sperner hypergraphs was studied. It was shown that such hypergraphs can be decomposed in a particular way and that they form a subclass of the class of threshold hypergraphs studied by Golumbic [31], by Reiterman et al. [56] and, in the equivalent context of threshold monotone Boolean functions, also by Muroga [48] and by Peled and Simeone [52]. (Precise definitions of all relevant concepts will be given in Section 2.)
While threshold hypergraphs are defined using the existence of certain weights, the Sperner property is a ‘purely combinatorial’ notion that forms a sufficient condition for thresholdness. Purely combinatorial necessary conditions for thresholdness are also known and follow from the characterization of threshold Boolean functions due to Chow [14] and Elgot [25]. The condition states that a hypergraph is threshold if and only if a certain obvious necessary combinatorial condition, called asummability, for all , is satisfied. For certain families of hypergraphs derived from graphs asummability implies Spernerness, which means that in such cases all the three (generally properly nested) properties of Spernerness, thresholdness, and asummability coincide. Early applications of this idea are implicit in the works of Chvátal and Hammer [15] and Benzaken and Hammer [2] characterizing threshold and domishold graphs, respectively (see Section 3
for details). By definition, these are graphs that admit a nonnegative linear vertex weight function separating the characteristic vectors of all independent sets, resp. dominating sets from the characteristic vectors of all other sets. The corresponding hypergraphs derived from a given graph, obtained using the notion of transversal hypergraphs, are the hypergraphs of inclusionminimal vertex covers and inclusionminimal closed neighborhoods, respectively.
More recent examples involve the works of Chiarelli and Milanič [12, 11] who made use of a slightly more general class than Sperner hypergraphs, called dually Sperner hypergraphs, to characterize two classes of graphs defined by the following properties: every induced subgraph has a nonnegative linear vertex weight function separating the characteristic vectors of all total dominating sets [11], resp. connected dominating sets [12], from the characteristic vectors of all other sets. In these cases, the corresponding transversal hypergraphs derived from a graph are the hypergraphs of inclusionminimal (open) neighborhoods and inclusionminimal cutsets, respectively. Due to the close relation between Sperner and dually Sperner hypergraphs (see [9]), all the results from [12, 11] can be equivalently stated using Sperner hypergraphs. In particular, the results of the extended abstract [11] are stated using the Sperner property in the full version of the paper [13].
A common approach in these studies is to consider various hypergraphs associated to a given graph and investigate the class of graphs for which the associated hypergraph is threshold. The goal is then to characterize these graph classes and, in particular, to understand under what conditions they are hereditary (that is, it is closed under vertex deletion). In some of the above cases (namely in the cases of threshold and domishold graphs) the resulting graph class is hereditary, while in the other two cases mentioned above the condition for induced subgraphs has to be imposed on the graph as part of the definition, since in general those properties are not hereditary.
Our results
We present several novel applications of the notion of Sperner hypergraphs and their structure to graphs. Our results can be summarized as follows.
1. Characterizations of threshold and domishold graphs. We show that forbidden induced subgraph characterizations of threshold and domishold graphs due to Chvátal and Hammer [15] and Benzaken and Hammer [2] imply that for several families of hypergraphs derived from graphs, some or all of the three (generally properly nested) properties of Spernerness, thresholdness, and asummability coincide. These families include the vertex cover, clique, closed neighborhood, and dominating set hypergraphs (see Theorems 3.2, 3.3, 3.5, and 3.6). For example, we show that in the class of clique hypergraphs of graphs, Spernerness, thresholdness, and asummability are all equivalent and that they characterize the threshold graphs. Furthermore, threshold graphs are exactly the cooccurrence graphs of 1Sperner hypergraphs (Theorem 3.4).
These results are interesting for multiple reasons. First, they give new characterizations of threshold graphs, which further extend the long list [45]. Second, our characterizations of clique hypergraphs of threshold graphs parallels the characterization of clique hypergraphs of chordal graphs due to Lauritzen et al. [41]. Third, our characterizations of threshold and domishold graphs implicitly address the question by Herranz [36] asking about further families of Boolean functions that enjoy the property that thresholdness and asummability coincide. Our results imply that this property is enjoyed by monotone Boolean functions corresponding to various hypergraphs naturally associated to graphs: vertex cover, clique, closed neighborhood, and dominating set hypergraphs. Finally, it is not at all a priori obvious whether the class of graphs whose clique hypergraphs are threshold is hereditary or not. Interestingly, it turns out that it is. Considering instead of the family of maximal cliques the families of edges (or, equivalently, the transversal family of inclusionminimal vertex covers) or of inclusionminimal dominating sets (or, equivalently, the transversal family of inclusionminimal closed neighborhoods) also results in hereditary graphs classes. In contrast, the inclusionminimal (open) neighborhoods or inclusionminimal cutsets do not define hereditary graph classes.
2. Decomposition results for four classes of graphs. We apply the decomposition theorem for Sperner hypergraphs to derive decomposition theorems for four interrelated classes of graphs: two subclasses of split graphs, a class of bipartite graphs, and a class of cobipartite graphs (Theorems 5.2, 5.3, 5.4, and 5.5, respectively). This is done by exploiting a variety of ways of how the incidence relation between the vertices and the hyperedges of a hypergraph can be represented with a graph, expressing the Sperner property of hypergraphs in terms of the corresponding derived graphs, and finally translating the decomposition of Sperner hypergraphs to decompositions of the corresponding graphs. These decompositions are recursive and are described in terms of matrix partitions of graphs, a concept introduced by Feder et al. [26] and studied in a series of papers (see, e.g., [34] and references therein).
3. New classes of graphs of bounded cliquewidth. We explore several consequences of the obtained graph decomposition results. The first ones are related to cliquewidth, a graph parameter introduced by Courcelle et al. [19], whose importance is mainly due to the fact that many hard decision and optimization problems are polynomially solvable in classes of graphs of uniformly bounded cliquewidth [55, 17, 50]. This makes it important to identify graph classes of bounded, resp. unbounded cliquewidth [38]. Each of the four considered graph classes obtained from Sperner hypergraphs is defined by forbidding one particular vertex graph as induced subgraph (one of the two graphs in Fig. 1 or their complements) and an extra condition on the structure of the neighborhoods.
As a consequence of the decomposition theorems, we show that a graph in any of the four classes has cliquewidth at most (Theorems 6.2, 6.3, 6.4, and 6.5). This result is in sharp contrast with the fact that the four graph classes defined by the same forbidden induced subgraphs but without any restrictions on the neighborhoods are of unbounded cliquewidth. In particular, this is the case for the class of free split graphs, where is one of the graphs depicted in Fig. 1.
Similar results, but incomparable to ours, were obtained for a class of bipartite graphs defined by two vertex forbidden induced subgraphs Fouquet et al. [28]; see also [43, 29, 61] for related work. Our results for bipartite graphs directly address the remark from [29]: “It is challenging to find new classes of bipartite graphs, defined both by their decomposition tree and another property (…) and to use this for new efficient algorithms design.” Naturally, our approach is applicable also to split and cobipartite graphs.
4. New polynomially solvable cases of variants of domination. Finally, we make further use of the bounded cliquewidth result by identifying new polynomially solvable cases of three basic variants of the dominating set problem. We show that the dominating set problem, the total dominating set problem, and the connected dominating set problem are all polynomialtime solvable in the class of free split graphs (Theorem 7.5). This result is sharp in the sense that all three problems are hard in the class of split graphs [40, 5, 16], as well as in the class of free graphs. The result for free graphs follows from the fact that the class of free graphs is a superclass of the class of line graphs, in which all these problems are known to be hard [64, 46, 35]. Moreover, strong inapproximability results are known for these variants of domination in the class of split graphs: for every , the dominating set, the total dominating set, and the connected dominating set problem cannot be approximated to within a factor on in the class of vertex split graphs, unless . Theorem 6.5 in [8] provides this result for domination and total domination, while for connected domination the same result follows from the fact that all the three problems are equivalent in any class of split graphs (see Lemma 7.2).
Structure of the paper
In Section 2, we collect the necessary preliminaries on graphs and hypergraphs, including a decomposition theorem for Sperner hypergraphs. In Section 3, we consider the characterizations of threshold and domishold graphs due to Chvátal and Hammer [15] and Benzaken and Hammer [2] and use them to obtain further characterizations of these classes in terms of Spernerness, thresholdness, and asummability properties of their vertex cover, clique, dominating set, and closed neighborhood hypergraphs. In Section 4, we associate various types of incidence graphs to a hypergraph and translate the Spernerness property of the hypregraph to the incidence graphs. In Section 5, we apply the decomposition theorem for Sperner hypergraphs to derive decomposition theorems for four classes of graphs. These theorems are then used in Section 6 to infer that each of the graph in resulting four classes has cliquewidth at most . Finally, in Section 7 we study the consequences of our structural approach for domination problems in a subclass of split graphs.
2 Preliminaries
2.1 Preliminaries on graphs
All graphs in this paper will be finite, simple, and undirected. Let be a graph. Given a vertex , its neighborhood is the set, denoted by , of vertices adjacent to , and its closed neighborhood is the set, denoted by , of vertices adjacent or equal to . An independent set (also called a stable set) in is a set of pairwise nonadjacent vertices, and a clique is a set of pairwise adjacent vertices. A dominating set in is a set such that every vertex of is either in or has a neighbor in . A total dominating set in is a set such that every vertex of has a neighbor in . A connected dominating set in is a dominating set that induces a connected subgraph. Dominating Set, Total Dominating Set, and Connected Dominating Set are the problems of finding a minimum dominating set, a total dominating set, resp., connected dominating set in a given graph. (Note that a graph has a total dominating set if and only if it has no isolated vertices and it has a connected dominating set if and only if it is connected.) We denote the minimum size of a dominating set, total dominating set, and connected dominating set in a graph by , , and , respectively.
Given a set of graphs and a graph , we say that is free if and only if has no induced subgraph isomorphic to a member of . Moreover, for a graph , we say that is free if it is free. Given a graph and a vertex , we denote by the subgraph of induced by . We denote the disjoint union of two graphs and by ; in particular, we write for the disjoint union of two copies of . As usual, we denote by , , and the path, the cycle, and the complete graph with vertices, respectively.
Split, bipartite, and cobipartite graphs. A graph is split if its vertex set can be partitioned into a (possibly empty) clique and a (possibly empty) independent set. A split partition of a split graph is a pair such that is a clique, is an independent set, , and . A graph is bipartite (or, shortly, a bigraph) if its vertex set can be partitioned into two (possibly empty) independent sets. A bipartition of a bipartite graph is a pair of two independent sets in such that and . A graph is cobipartite if its complement , that is, the graph with vertex set in which two distinct vertices are adjacent if and only if they are nonadjacent in , is bipartite.
Threshold graphs. A graph is said to be threshold if there exist a nonnegative integer weight function and a nonnegative integer threshold such that for every subset , we have if and only if is an independent set. Threshold graphs were introduced by Chvátal and Hammer in 1970s [15] and were afterwards studied in numerous papers. Many results on threshold graphs are summarized in the monograph by Mahadev and Peled [45].
We now recall three of the many characterizations of threshold graphs. By we denote the vertex path, by the vertex cycle, and by the disjoint union of two copies of (the complete graph of order two).
Theorem 2.1 (Chvátal and Hammer [15]).
A graph is threshold if and only if is free.
Theorem 2.2 (Chvátal and Hammer [15]).
A graph is threshold if and only if is split with a split partition such that there exists an ordering of such that for all .
The join of two vertexdisjoint graphs and is the graph, denoted by , obtained from the disjoint union by adding to it all edges with one endpoint in and one endpoint in .
Corollary 2.3.
A graph is threshold if and only if can be built from the vertex graph by an iterative application of the operations of adding an isolated vertex (that is, disjoint union with ) or a universal vertex (that is, join with ).
Domishold graphs. A graph is said to be domishold if there exist a nonnegative integer weight function and a nonnegative integer threshold such that for every subset , we have if and only if is a dominating set. Domishold graphs were introduced by Benzaken and Hammer [2], who characterized them with the set of forbidden induced subgraphs depicted in Fig. 2.
Theorem 2.4 (Benzaken and Hammer [2]).
A graph is domishold if and only if is free.
Matrix partitions. Let be a symmetric matrix with entries in . An partition of a graph is a partition of the vertex set of into parts such that two distinct vertices in parts and (possibly with ) are adjacent if and nonadjacent if . In particular, for these restrictions mean that is either a clique, or an independent set, or is unrestricted, when is , or , or , respectively. In general, some of the parts may be empty.
Cliquewidth. A graph is a graph whose vertices are equipped with labels from the set . The cliquewidth of a graph , denoted , is the smallest positive integer such that there exists a graph isomorphic to that can be obtained as the result of a finite sequence of iteratively applying the following operations:

where : creation of a vertex graph with its unique vertex labeled ,

, disjoint union of two graphs and ,

where with : the graph obtained from the graph by assigning label to all vertices having label

where with : the graph obtained from the graph by adding to it all edges of the form where has label and has label .
An algebraic expression describing the above construction is called a expression of . For example, the following is a expression of a vertex path with vertices and edges :
Given a expression of , we denote by its length, which is the number of symbols one needs to write it down.
2.2 Preliminaries on hypergraphs
A hypergraph is a pair where is a finite set of vertices and is a set of subsets of , called hyperedges. A hypergraph is uniform if all its hyperedges have size . In particular, graphs are precisely the uniform hypergraphs. Every hypergraph with a fixed pair of orderings of its vertices and edges, say , and , can be represented with its incidence matrix having rows and columns indexed by edges and vertices of , respectively, and being defined as
Sperner hypergraphs. A hypergraph is said to be Sperner if no hyperedge contains another one, that is, if and implies ; see, e.g., Sperner [59], Shapiro [58], Berge and Duchet [3]. Sperner hypergraphs were studied in the literature under different names including simple hypergraphs by Berge [4], clutters by Billera [7, 6] and by Edmonds and Fulkerson [30, 24], and coalitions
in the game theory literature
[62].Dually Sperner hypergraphs. Sperner hypergraphs can be equivalently defined as the hypergraphs such that every two distinct hyperedges and satisfy . This point of view motivated Chiarelli and Milanič to define in [12] a hypergraph to be dually Sperner if every two distinct hyperedges and satisfy . It was shown in [12] that dually Sperner hypergraphs are threshold.
1Sperner hypergraphs. Following Boros et al. [9], we say that a hypergraph is Sperner if every two distinct hyperedges and satisfy
In particular, is Sperner if every two distinct hyperedges and satisfy
or, equivalently, if, for any two distinct hyperedges and of with , we have .
As proved by Boros et al. [9], Sperner hypergraphs admit a recursive decomposition. In order to state the result, we first need to recall some definitions. The following operation produces (with one exception) a new Sperner hypergraph from a given pair of Sperner hypergraphs.
Definition 2.5 (Gluing of two hypergraphs).
Given a pair of vertexdisjoint hypergraphs and and a new vertex , the gluing of and is the hypergraph such that
and
Let us illustrate the operation of gluing on an example, in terms of incidence matrices. Let and for , and let us denote by , resp. , the matrix of all zeroes, resp. of all ones. Then, the incidence matrix of the gluing of and can be written as
See Fig. 3 for an example.
Observation 2.6.
If the gluing of and is a Sperner hypergraph, then and are also Sperner.
Given a vertex of a hypergraph , we say that a hypergraph is decomposable if for every two hyperedges such that , we have . Equivalently, is decomposable if its vertex set can be partitioned as such that for some hypergraphs and .
The following is the main structural result about Sperner hypergraphs.
Theorem 2.7 (Boros et al. [9]).
Every Sperner hypergraph with is decomposable for some , that is, it is the gluing of two Sperner hypergraphs.
Threshold hypergraphs. A hypergraph is said to be threshold if there exist a nonnegative integer weight function and a nonnegative integer threshold such that for every subset , we have if and only if for some .
Threshold hypergraphs were defined in the uniform case by Golumbic [31] and studied further by Reiterman et al. [56]. 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, see, e.g., [48]. A polynomialtime recognition algorithm for threshold monotone Boolean functions represented by their complete DNF was given by Peled and Simeone [52]
. The algorithm is based on linear programming and implies a polynomialtime recognition algorithm for threshold hypergraphs.
The mapping that takes every hyperedge to its characteristic vector , defined by
shows that the sets of hyperedges of threshold Sperner hypergraphs are in a onetoone correspondence with the sets of minimal feasible binary solutions of the linear inequality . A set of vertices 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 independent sets from the characteristic vectors of dependent sets.
For later use, we recall the following theorem.
Theorem 2.8 was derived in [9] from Theorem 2.7. However, note that every Sperner hypergraph is dually Sperner, which means that Theorem 2.8 also follows from the fact that dually Sperner hypergraphs are threshold, which was established earlier than Theorem 2.7 but in a nonconstructive way [12].
asummable hypergraphs. A hypergraph is asummable if it has no (not necessarily distinct) independent sets and (not necessarily distinct) dependent sets such that
A hypergraph is asummable if it is asummable for every .
The result was given in terms of monotone Boolean functions; more recent and complete information on this can be found in [21]. For later use we state explicitly the simple fact that asummability is a necessary condition for thresholdness.
Corollary 2.10.
Every threshold hypergraph is asummable.
It is also known that asummability does not imply thresholdness in general [21], and thresholdness does not imply Spernerness (as can be seen by considering the edge set of the complete graph ).
Transversal hypergraphs. Let be a hypergraph. A transversal of is a set of vertices intersecting all hyperedges of . The transversal hypergraph is the hypergraph with vertex set in which a set is a hyperedge if and only if is an inclusionminimal transversal of .
Observation 2.11 (see, e.g., Berge [4]).
If is a Sperner hypergraph, then .
A pair of a mutually transversal Sperner hypergraphs naturally corresponds to a pair of dual monotone Boolean functions, see [21].
Observation 2.12 (see, e.g., Muroga [48]).
For every integer , a Sperner hypergraph is asummable if and only if its transversal hypergraph is asummable.
Corollary 2.13 (see, e.g., Crama and Hammer [21]).
For every integer , a Sperner hypergraph is threshold if and only if its transversal hypergraph is threshold.
Clique hypergraphs and conformal hypergraphs. Given a graph , the clique hypergraph of is the hypergraph with vertex set in which the hyperedges are exactly the (inclusion)maximal cliques of . A hypergraph is is conformal if for every set such that every pair of elements in is contained in a hyperedge, there exists a hyperedge containing .
Theorem 2.14 (Berge [4]).
A Sperner hypergraph is conformal if and only if it is the clique hypergraph of a graph.
Interrelations between the considered classes of graphs and hypergraphs. In Fig. 4, we show the Hasse diagram of the partial order of the hypergraph classes mentioned above, along with some classes of graphs, ordered with respect to inclusion. The diagram includes equalities between the following classes of hypergraphs:
The fact that every threshold graph is split follows from Theorem 2.2, while the fact that every dually Sperner hypergraph is threshold was proved by Chiarelli and Milanič [12]. The remaining inclusions are trivial or follow by transitivity.
Finally, the following examples show that all inclusions are strict and that there are no other inclusions (in particular, there are no other equalities):

the complete graph is a uniform Sperner hypergraph that it is not conformal;

the complete graph is a threshold graph and, when viewed as a uniform hypergraph, it is threshold and Sperner but neither dually Sperner nor conformal;

the path is a split graph but, as a uniform hypergraph, it is conformal and Sperner but not asummable;

the clique hypergraph of the complete graph (which is threshold) is not uniform;

the hypergraph with vertex set and hyperedge set is dually Sperner but not Sperner,

the clique hypergraph of the graph is conformal and Sperner but not Sperner;
The remaining noninclusions follow by transitivity.
3 Characterizations of threshold and domishold graphs
In this section, we consider the characterizations of threshold and domishold graphs due to Chvátal and Hammer [15] and Benzaken and Hammer [2] and show that they imply that for several families of hypergraphs derived from graphs, some or all of the three (generally properly nested) properties of Spernerness, thresholdness, and asummability coincide.
3.1 Threshold graphs
The first result along these lines follows easily from the characterizations of threshold graphs by forbidden induced subgraphs due to Chvátal and Hammer, see e.g., Corollary 1D [15].
Theorem 3.1 (Chvátal and Hammer [15]).
For a graph , the following statements are equivalent:

is a threshold graph.

is a (uniform) threshold hypergraph.

is a (uniform) asummable hypergraph.
To see that the list above cannot be extended with the property of being Sperner, consider, for example, the graph . This is a threshold graph but it is not Sperner.
Threshold graphs have many other characterizations (see, e.g., [45]). We now prove several more, which can also be seen as characterizations of certain classes of hypergraphs.
A vertex cover of a graph is a set such that every edge of has at least one endpoint in . Given a graph , the vertex cover hypergraph is the hypergraph with vertex set in which a set is a hyperedge if and only if is an inclusionminimal vertex cover. In contrast to Theorem 3.1, if we focus on characterizing thresholdness in terms of the vertex cover hypergraph, then the property of Spernerness appears.
Theorem 3.2.
For a graph , the following statements are equivalent:

is threshold.

The vertex cover hypergraph is Sperner.

The vertex cover hypergraph is threshold.

The vertex cover hypergraph is asummable.
Proof.
The chain of implications follows from Theorem 2.8 and Corollary 2.10. Since the vertex cover hypergraph is Sperner and it is the transversal hypergraph of the uniform hypergraph , creftypecap 2.11 implies that . Therefore, by creftypecap 2.12, is asummable if and only if is asummable, which together with Theorem 3.1 yields .
For the implication , suppose that the vertex cover hypergraph is not Sperner. We will prove that is not threshold by showing that contains an induced subgraph isomorphic to , , or . Since is Sperner but not Sperner, there exist two minimal vertex covers and of such that . Since is a vertex cover, the set is independent. Similarly, is independent. We claim that the bipartite subgraph of induced by has no isolated vertices. By symmetry it suffices to show that every vertex has a neighbor in . If this were not the case, then by the minimality of we infer that would have a neighbor in . However, this would contradict the fact that is a vertex cover. Since every bipartite graph without isolated vertices and having at least two vertices in both sides of a bipartition has an induced , , or , we conclude that is not threshold by Theorem 2.1. ∎
We show next that in the class of conformal Sperner hypergraphs, the notion of thresholdness coincides with both asummability and Spernerness. Moreover, it exactly characterizes threshold graphs. Recall that a Sperner hypergraph is conformal if and only if it is the clique hypergraph of a graph (Theorem 2.14).
Theorem 3.3.
For a graph , the following statements are equivalent:

is threshold.

The clique hypergraph is Sperner.

The clique hypergraph is threshold.

The clique hypergraph is asummable.
Proof.
For the implication , suppose that the clique hypergraph is not Sperner. Since is Sperner but not Sperner, there exist two maximal cliques and of such that . Clearly, and are cliques. By the maximality of , every vertex in has a nonneighbor in . Similarly, every vertex in has a nonneighbor in . Thus, the complement of the subgraph of induced by is a bipartite graph with at least two vertices in both sides and without isolated vertices. Therefore, it contains an induced , , or , and since this family of graphs is closed under complementation, we conclude that is not threshold by Theorem 2.1.
The chain of implications follows from Theorem 2.8 and Corollary 2.10.
Now we prove the implication . We break the proof into a small series of claims.
Claim 1.
For any graph , if , , and such that , then the set is an independent set of the hypergraph .
Proof.
If is a clique of , then is also a clique of , and hence is not maximal. ∎
Claim 2.
If is asummable, and , , and then .
Proof.
Claim 3.
If is asummable, and , , and , then the set is an independent set of the hypergraph .
Proof.
If is a maximal clique of , then

is not a clique, and hence there exists a vertex (such that );

is not a clique, and hence there exists a vertex (such that ).
Consequently we have vertices and such that implying . This contradicts Claim 2, which proves that such a maximal clique cannot exist. ∎
Claim 4.
If is asummable then is a threshold graph.
Proof.
Let us recall the forbidden subgraph characterization of threshold graphs by Theorem 2.1, and assume indirectly that there exist four distinct vertices such that and (that is, that induces either a , a , or a ). Let us then consider maximal cliques and of . Since such maximal cliques do exist.
Our assumptions that imply that and .
Then we can apply Claim 3 and obtain that both and are independent sets of the hypergraph . Since , it also follows that is an independent set of . Consequently, the equality
contradicts the asummability of . This contradiction proves that such a set of four vertices cannot exists, from which the claim follows by Theorem 2.1. ∎
The last claim proves the implication , completing the proof of the theorem. ∎
Remark 1.
The independent set hypergraph of a graph is the hypergraph with vertex set and in which the hyperedges are exactly the maximal independent sets of . Since the class of threshold graphs is closed under taking complements, one could obtain three further characterizations of threshold graphs by replacing the clique hypergraph with the independent set hypergraph in any of the properties – in Theorem 3.3.
Let us finally show that the property of Spernerness is related to the family of threshold graphs also in the following way. Given a hypergraph , its cooccurence graph is the graph with vertex set in which two vertices are adjacent if and only if there exists a hyperedge of containing both (see, e.g., [21, Chapter 10]).
Theorem 3.4.
A graph is threshold if and only if it is the cooccurence graph of some Sperner hypergraph.
Proof.
If is threshold, then by Theorem 3.3 the clique hypergraph is Sperner. Since two vertices of are adjacent if and only if they belong to some maximal clique, the cooccurrence graph of is itself. This shows the forward implication.
Suppose now that is a Sperner hypergraph and let be its cooccurence graph. We prove that is threshold using induction on . For , the statement is trivial, so let us assume . We may also assume that , since otherwise is edgeless and thus threshold. If has an isolated vertex, then by deleting it we get again a Sperner hypergraph. Furthermore, to the cooccurence graph of the resulting hypergraph we can add back an isolated vertex to obtain . Since by Corollary 2.3, this operations preserves thresholdness, we may assume that has no isolated vertices.
Since is a Sperner hypergraph with , Theorem 2.7 implies the existence of a vertex such that is decomposable. Let denote such a vertex and let be the corresponding gluing of and . That is, and . We may assume without loss of generality that has no isolated vertex. For , let be the cooccurrence graph of . By the induction hypothesis, and are threshold graphs (note that this is the case also if some of the sets is empty).
Suppose first that . In this case, and since has no isolated vertices, . Therefore, is the graph obtained from by adding to it the universal vertex . Thus, by Corollary 2.3, is threshold.
Assume finally that . Since has no isolated vertices, . It is now not difficult to see that we can obtain in two steps: first, we add to vertex as an isolated vertex; second, to the so obtained graph we add the vertices of (one by one, in any order) as universal vertices. Since is threshold, it follows from Corollary 2.3 that is threshold. This completes the proof. ∎
3.2 Domishold graphs
Similar characterizations as for threshold graphs can be derived for domishold graphs. Note that in the proof of Theorem 3.2 we used the fact that the vertex cover hypergraph is transversal to the family of edges of the graph. In the case of domishold graphs, we first consider the hypergraph transversal to the family of (inclusionminimal) dominating sets. Given a graph , the closed neighborhood hypergraph is the hypergraph with vertex set in which a set is a hyperedge if and only if is an inclusionminimal set of the form for some .
Theorem 3.5.
For a graph , the following statements are equivalent:

is domishold.

The closed neighborhood hypergraph is Sperner.

The closed neighborhood hypergraph is threshold.

The closed neighborhood hypergraph is asummable.
Proof.
For the implication , suppose that the closed neighborhood hypergraph is not Sperner. Since is Sperner but not Sperner, there exist two vertices such that . Clearly, . We consider two cases depending on whether and are adjacent or not.
Suppose first that . Then there exist vertices such that and . Then, must induce either a (if and are adjacent) or a (otherwise).
Suppose now that . Then there exist four distinct vertices in such that and . If does contains an induced , then is not domishold by Theorem 2.4. We may therefore assume that does not contain an induced . This implies that every vertex of is adjacent to every vertex of . Since is adjacent to neither nor and is adjacent to neither nor , the subgraph of induced by these six vertices is isomorphic to one of , , (see Fig. 2).
In both cases, we conclude that is not domishold by Theorem 2.4.
The chain of implications follows from Theorem 2.8 and Corollary 2.10.
To prove the implication , we prove the contrapositive, that is, if is not domishold, then is not a asummable hypergraph. Suppose that is not domishold. By Theorem 2.4 has an induced subgraph isomorphic to a graph from the set (see Fig. 2).
Suppose first that is isomorphic to or to . Then there exist four distinct vertices such that and . Consider the sets , , , and . It is clear that and are dependent sets of . The sets and , however, are independent. By symmetry, it suffices to show that is independent in , that is, that it contains no closed neighborhood of . We have since . Furthermore, for every we have since . Finally, for every we have since . Therefore, is independent. But now, since we have , is not asummable, which is what we wanted to prove.
Suppose now that is isomorphic to , , or . We may assume that is free, since otherwise can repeat the arguments above. The fact that is an induced subgraph of implies the existence of a set of six distinct vertices such that every vertex from is adjacent to every vertex from , vertex is not adjacent to or , and vertex is not adjacent to and , see Fig. 5.
Consider the sets , , , and . It is clear that and are dependent sets of . Since , we can infer that is not asummable (which is what we want to prove). if the sets and are independent. So we may assume that , say, is not independent in . This means that there exists a vertex such that . Then since , and similarly . Since and every vertex in is either or belongs to , we infer that is adjacent to . However, since , vertices and are not adjacent to . But now, the vertex set induces a in , a contradiction. ∎
Given a graph , the dominating set hypergraph is the hypergraph with vertex set in which a set is a hyperedge if and only if is an inclusionminimal dominating set of . The analogue of Theorem 3.1 for domishold graphs is the following.
Theorem 3.6.
For a graph , the following statements are equivalent:

is domishold.

The dominating set hypergraph is threshold.

The dominating set hypergraph is asummable.
Proof.
Since the dominating set hypergraph is Sperner and it is the transversal hypergraph of the closed neighborhood hypergraph , creftypecap 2.11 implies that . Therefore, by Corollary 2.13 and creftypecap 2.12, the dominating set hypergraph is threshold, resp. asummable, if and only if the closed neighborhood hypergraph is threshold, resp. asummable. The theorem now follows from Theorem 3.5. ∎
To see that the list from Theorem 3.6 cannot be extended with the property of the dominating set hypergraph being Sperner, consider, for example, the graph . This is a domishold graph but its dominating set hypergraph is isomorphic to , which is not Sperner.
3.3 Total domishold and connecteddomishold graphs
We conclude this section by stating two related theorems due to Chiarelli and Milanič [12, 13] about the structure of graphs defined similarly as domishold graphs, but with respect to total, resp. connected domination. A graph is said to be total domishold (resp., connecteddomishold) if there exist a nonnegative integer weight function and a nonnegative integer threshold such that for every subset , we have if and only if is a total dominating set (resp., a connected dominating set). Given a graph , the neighborhood hypergraph is the hypergraph with vertex set in which a set is a hyperedge if and only if is an inclusionminimal set of the form for some . A cutset in a graph is a set such that is disconnected. The cutset hypergraph is the hypergraph with vertex set in which a set is a hyperedge if and only if is an inclusionminimal cutset in .
Theorem 3.7 (Chiarelli and Milanič [12]).
For a graph , the following statements are equivalent:

Every induced subgraph of is total domishold.

The neighborhood hypergraph of every induced subgraph of is Sperner.

The neighborhood hypergraph of every induced subgraph of is threshold.

The neighborhood hypergraph of every induced subgraph of is asummable.
Theorem 3.8 (Chiarelli and Milanič [13]).
For a graph , the following statements are equivalent:

Every induced subgraph of is connecteddomishold.

The cutset hypergraph of every induced subgraph of is Sperner.

The cutset hypergraph of every induced subgraph of is threshold.

The cutset hypergraph of every induced subgraph of is asummable.
4 Bipartite and split representations of Sperner hypergraphs
In the remainder of the paper, we apply the decomposition theorem for Sperner hypergraphs (Theorem 2.7) to derive decomposition theorems for four classes of graphs and derive some consequences. The four classes come in two pairs: a class of split graphs and their complements (which are also split), and a class of bipartite graphs and their complements (which are cobipartite). For our purpose it will be convenient to consider split graphs as already equipped with a split partition and bipartite graphs (bigraphs) as already equipped with a bipartition. Partly following the terminology of Dabrowski and Paulusma [22], we will say that a labeled split graph is a triple such that is a split graph with a split partition . Similarly, a labeled bigraph is a triple such that is a bigraph with a bipartition . Given two labeled split graphs and , we say that is (isomorphic to) an induced subgraph of if there exists a pair of injective mappings and such that for all , vertices and are adjacent in if and only if vertices and are adjacent in . If is not an induced subgraph of , we say that is free. For labeled bigraphs, the definitions are similar.
To each hypergraph we can associate three types of incidence graphs in a natural way: one labeled bipartite graph and two labeled split graphs.
Definition 4.1.
Given a hypergraph , we define:

the bigraph of as the labeled bigraph such that is adjacent to if and only if ,

the vertexclique split graph of as the labeled split graph such that is a clique and is adjacent to if and only if , and

the edgeclique split graph of as the labeled split graph such is a clique and is adjacent to if and only if .
See Fig. 6 for an example.
Representing a hypergraph with its bigraph is standard in studies of general [4] or highly regular [42, 20, 54] hypergraphs. Representing a hypergraph with a split incidence graph has also been used in the literature, e.g., by Kanté et al. [39], followed by Chiarelli and Milanič [12, 13], who considered the vertexclique split graph (using slightly different terminology).
We will characterize Sperner hypegraphs in terms of properties of their bigraphs and split graphs. Recall that a hypergraph is said to be dually Sperner if every two distinct hyperedges and satisfy , and that is Sperner if and only if it is Sperner and dually Sperner. Both the Sperner and the dually Sperner property can be easily translated to properties of their incidence graphs. The corresponding characterizations of Sperner hypegraphs will then follow as corollaries.
The Sperner property is naturally expressed in terms of constraints on the neighborhoods of vertices in one of the parts of the bipartition of a labeled bigraph (resp., split partition of a labeled split graph) intersected with the other part.
Definition 4.2.
A labeled bigraph is said to be rightSperner if for all , if then . A labeled split graph is said to be cliqueSperner if for all , if then , and independentSperner if for all , if then .
Observation 4.3.
For every hypergraph , the following conditions are equivalent:

is Sperner.

The bigraph of is rightSperner.

The vertexclique split graph of is independentSperner.

The edgeclique split graph of is cliqueSperner.
To express the dually Sperner property of a hypergraph in terms of properties of its incidence graphs, we introduce three small graphs, namely the , the graph denoted by and obtained from by adding the edge between the two vertices of degree two, and by the (graph theoretic) complement of . When necessary, we will consider the bigraph as a labeled bigraph, with respect to the bipartition shown in Fig. 7, and the split graphs and as labeled split graphs, with respect to their unique split partitions.
Remark 2.
When the bigraph and the labeled split graphs and are considered labeled, as shown in Fig. 8, they are the bigraph, the edgeclique split graph, and the vertexclique split graph, respectively, of the smallest nonSperner hypergraph.
The dually Sperner property of a hypergraph can be expressed in terms of properties of its incidence graphs as follows.
Observation 4.4.
For every hypergraph , the following conditions are equivalent:

is dually Sperner.

The bigraph of is free in the labeled sense.

The vertexclique split graph of is free.

The edgeclique split graph of is free.
Expressing the Sperner property of a hypergraph in terms of properties of its incidence graphs now follows.
Corollary 4.5.
For every hypergraph , the following conditions are equivalent:

is Sperner.

The bigraph of is rightSperner and free in the labeled sense.

The vertexclique split graph of is independentSperner and free.

The edgeclique split graph of is cliqueSperner and free.
5 Decomposition theorems for four graph classes
We will now apply the connection between Sperner hypergraphs and their incidence graphs explained in Section 4 to derive decomposition theorems for a class of split graphs and their complements, and a class of bigraphs and their complements. To this end, we need to generalize Definition 4.2 to general (not labeled) bigraphs and split graphs. A bigraph is said to be rightSperner if it has a bipartition such that the labeled bigraph is rightSperner. A split graph is said to be cliqueSperner (resp., independentSperner) if it has a split partition such that the labeled split graph is cliqueSperner (resp., independentSperner).
We first prove the decomposition theorem for the class of free cliqueSperner split graphs. We start with a useful lemma.
Lemma 5.1.
Suppose that a split graph has a split partition such that the labeled split graph is cliqueSperner. Then, the partition has the following properties:

if is edgeless, then ,

if , then for some nonisolated vertex , and

if has at least two edges, then is the unique split partition of such that is a maximal independent set in .
Furthermore, there is an
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