# Characterizing and decomposing classes of threshold, split, and bipartite graphs via 1-Sperner hypergraphs

A hypergraph H is said to be 1-Sperner if for every two hyperedges the smallest of their two set differences is of size one. The authors introduced this concept in 2015 and completely characterized this class of hypergraphs via a decomposition theorem. Here, we present several applications of 1-Sperner hypergraphs and their structure to graphs. In particular, we consider the classical characterizations of threshold and domishold graphs and use them to obtain further characterizations of these classes in terms of 1-Spernerness, thresholdness, and 2-asummability of their vertex cover, clique, dominating set, and closed neighborhood hypergraphs. Furthermore, we apply the decomposition theorem for 1-Sperner hypergraphs to derive decomposition theorems for 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, giving rise to new classes of graphs of bounded clique-width and new polynomially solvable cases of several domination problems.

## Authors

• 9 publications
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• ### Ergodic Theorems for PSPACE functions and their converses

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## 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 non-negative 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 inclusion-minimal vertex covers and inclusion-minimal 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 non-negative 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 inclusion-minimal (open) neighborhoods and inclusion-minimal 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 3.23.33.5, 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 co-occurrence graphs of 1-Sperner 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 inclusion-minimal vertex covers) or of inclusion-minimal dominating sets (or, equivalently, the transversal family of inclusion-minimal closed neighborhoods) also results in hereditary graphs classes. In contrast, the inclusion-minimal (open) neighborhoods or inclusion-minimal 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 5.25.3,  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 clique-width. We explore several consequences of the obtained graph decomposition results. The first ones are related to clique-width, 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 clique-width [55, 17, 50]. This makes it important to identify graph classes of bounded, resp. unbounded clique-width [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 clique-width at most (Theorems 6.26.36.4, 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 clique-width. 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 clique-width 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 polynomial-time 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 clique-width 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 non-adjacent 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 non-adjacent in , is bipartite.

Threshold graphs. A graph is said to be threshold if there exist a non-negative integer weight function and a non-negative 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 vertex-disjoint 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 non-negative integer weight function and a non-negative 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.

Clique-width. A -graph is a graph whose vertices are equipped with labels from the set . The clique-width 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 :

 η2,3(ρ3→2(ρ2→1(η2,3(η1,2(1(v1)⊕2(v2))⊕3(v3))))⊕3(v3)).

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

 AHi,j={1,if vj∈ei;0,otherwise.

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.

1-Sperner hypergraphs. Following Boros et al. [9], we say that a hypergraph is -Sperner if every two distinct hyperedges and satisfy

 1≤min{|e∖f|,|f∖e|}≤k.

In particular, is -Sperner if every two distinct hyperedges and satisfy

 min{|e∖f|,|f∖e|}=1,

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 vertex-disjoint hypergraphs and and a new vertex , the gluing of and is the hypergraph such that

 V(H)=V1∪V2∪{z}

and

 E(H)={{z}∪e∣e∈E1}∪{V1∪e∣e∈E2}.

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

 AH1⊙H2=(1m1,1AH10m1,n20m2,11m2,n1AH2).

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 non-negative integer weight function and a non-negative 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 polynomial-time 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 polynomial-time recognition algorithm for threshold hypergraphs.

The mapping that takes every hyperedge to its characteristic vector , defined by

 χev={1,if v∈e0,otherwise\,,

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 . 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 (Boros et al. [9], Chiarelli and Milanič [12]).

Every -Sperner hypergraph is threshold.

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 non-constructive 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

 k∑i=1χAi=k∑i=1χBi.

A hypergraph is asummable if it is -asummable for every .

###### Theorem 2.9 (Chow [14] and Elgot [25]).

A hypergraph is threshold if and only if it is asummable.

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 inclusion-minimal 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:

• threshold and asummable (Theorem 2.9);

• -uniform threshold, -uniform -asummable, and threshold graphs (Theorem 3.1);

• conformal -Sperner, conformal Sperner threshold, conformal Sperner -asummable hypergraphs, and clique hypergraphs of threshold graphs (Theorems 2.14 and 3.3).

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;

• an example of a -asummable non-threshold hypergraph is given, e.g., in [21, 13].

The remaining non-inclusions 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:

1. is a threshold graph.

2. is a (-uniform) threshold hypergraph.

3. 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 inclusion-minimal 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:

1. is threshold.

2. The vertex cover hypergraph is -Sperner.

3. The vertex cover hypergraph is threshold.

4. 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:

1. is threshold.

2. The clique hypergraph is -Sperner.

3. The clique hypergraph is threshold.

4. 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 non-neighbor in . Similarly, every vertex in has a non-neighbor 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.

If then Claim 1 and the equality

 χA+χB = χ(A∖{a})∪{b}+χ(B∖{b})∪{a}

contradicts the -asummability of . ∎

###### 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

 χA+χB = χ(A∪B)∖{a,c}+χ(A∩B)∪{a,c}

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 co-occurence 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 co-occurence 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 co-occurrence graph of is itself. This shows the forward implication.

Suppose now that is a -Sperner hypergraph and let be its co-occurence 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 co-occurence 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 co-occurrence 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 (inclusion-minimal) 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 inclusion-minimal set of the form for some .

###### Theorem 3.5.

For a graph , the following statements are equivalent:

1. is domishold.

2. The closed neighborhood hypergraph is -Sperner.

3. The closed neighborhood hypergraph is threshold.

4. 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 inclusion-minimal 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:

1. is domishold.

2. The dominating set hypergraph is threshold.

3. 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 connected-domishold 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., connected-domishold) if there exist a non-negative integer weight function and a non-negative 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 inclusion-minimal 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 inclusion-minimal cutset in .

###### Theorem 3.7 (Chiarelli and Milanič [12]).

For a graph , the following statements are equivalent:

1. Every induced subgraph of is total domishold.

2. The neighborhood hypergraph of every induced subgraph of is -Sperner.

3. The neighborhood hypergraph of every induced subgraph of is threshold.

4. 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:

1. Every induced subgraph of is connected-domishold.

2. The cutset hypergraph of every induced subgraph of is -Sperner.

3. The cutset hypergraph of every induced subgraph of is threshold.

4. The cutset hypergraph of every induced subgraph of is -asummable.

Characterizations of graphs every induced subgraph of which is total domishold, resp. connected-domishold, in terms of forbidden induced subgraphs were also given in [12, 13]. We omit them here.

## 4 Bipartite and split representations of 1-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 vertex-clique split graph of as the labeled split graph such that is a clique and is adjacent to if and only if , and

• the edge-clique 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 vertex-clique 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 right-Sperner if for all , if then . A labeled split graph is said to be clique-Sperner if for all , if then , and independent-Sperner if for all , if then .

###### Observation 4.3.

For every hypergraph , the following conditions are equivalent:

1. is Sperner.

2. The bigraph of is right-Sperner.

3. The vertex-clique split graph of is independent-Sperner.

4. The edge-clique split graph of is clique-Sperner.

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 edge-clique split graph, and the vertex-clique split graph, respectively, of the smallest non--Sperner 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:

1. is dually Sperner.

2. The bigraph of is -free in the labeled sense.

3. The vertex-clique split graph of is -free.

4. The edge-clique 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:

1. is -Sperner.

2. The bigraph of is right-Sperner and -free in the labeled sense.

3. The vertex-clique split graph of is independent-Sperner and -free.

4. The edge-clique split graph of is clique-Sperner and -free.

###### Proof.

Immediate from Observations 4.3 and 4.4, using the fact that is -Sperner if and only if it is both Sperner and dually Sperner. ∎

## 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 right-Sperner if it has a bipartition such that the labeled bigraph is right-Sperner. A split graph is said to be clique-Sperner (resp., independent-Sperner) if it has a split partition such that the labeled split graph is clique-Sperner (resp., independent-Sperner).

We first prove the decomposition theorem for the class of -free clique-Sperner 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 clique-Sperner. Then, the partition has the following properties:

• if is edgeless, then ,

• if , then for some non-isolated 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