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Circular-arc hypergraphs: Rigidity via connectedness
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Circular-arc hypergraphs: Rigidity via connectedness Johannes Köbler, Sebastian Kuhnert, Oleg Verbitsky ∗,1 Humboldt-Universität zu Berlin, Institut für Informatik, Unter den Linden 6, 10099 Berlin, Germany
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Article history: Received 18 September 2015 Received in revised form 1 June 2016 Accepted 16 August 2016 Available online xxxx Keywords: Circular-arc hypergraphs Circular-ones property Intersection graphs Proper circular-arc graphs Unique representations Graph canonization
abstract A circular-arc hypergraph H is a hypergraph admitting an arc ordering, that is, a circular ordering of the vertex set V (H ) such that every hyperedge is an arc of consecutive vertices. We give a criterion for the uniqueness of an arc ordering in terms of connectedness properties of H . This generalizes the relationship between rigidity and connectedness disclosed by Chen and Yesha (1991) in the case of interval hypergraphs. Moreover, we state sufficient conditions for the uniqueness of tight arc orderings where, for any two hyperedges A and B such that A ⊆ B ̸= V (H ), the corresponding arcs must share a common endpoint. We notice that these conditions are obeyed for the closed neighborhood hypergraphs of proper circular-arc graphs, implying for them the known rigidity results that were originally obtained using the theory of local tournament graph orientations. © 2016 Elsevier B.V. All rights reserved.
1. Introduction 1.1. Interval and circular-arc hypergraphs An interval ordering of a hypergraph H with a finite vertex set V = V (H ) is a linear ordering v1 , . . . , vn of V such that every hyperedge of H is an interval of consecutive vertices. This notion can be generalized to an arc ordering v1 , . . . , vn where the vertices are circularly ordered (i.e., v1 succeeds vn ) so that every hyperedge is an arc of consecutive vertices. An interval hypergraph is a hypergraph admitting an interval ordering. Similarly, if a hypergraph admits an arc ordering, we call it circular-arc (using also the shorthand CA). In the terminology stemming from computational genomics, interval hypergraphs are exactly those hypergraphs whose incidence matrix has the consecutive ones property; see, e.g., [7]. Similarly, a hypergraph is CA exactly when its incidence matrix has the circular ones property; see [8,18] for the relevance to computational genomics and [10,11] for the algorithmic aspects. Our goal is to study the conditions under which interval and circular-arc hypergraphs are rigid in the sense that they have a unique interval or arc ordering, respectively. Since any interval (or arc) ordering can be changed to another interval (or arc) ordering by reversing, we always mean uniqueness up to reversal. An obvious necessary condition for being rigid is that a hypergraph has no twins, that is, no two vertices such that every hyperedge contains either both or none of them. We say that two sets A and B overlap and write A G B, if A and B have nonempty intersection and neither of the two sets includes the other. To facilitate notation, we use the same character H to denote a hypergraph and the set of its hyperedges. We call H overlap-connected if the graph (H , G) is connected. A vertex of H is isolated if it is not contained in any hyperedge. As a starting point, we refer to the following rigidity result.
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Corresponding author. E-mail address: [email protected] (O. Verbitsky).
1 On leave from the Institute for Applied Problems of Mechanics and Mathematics, Lviv, Ukraine. http://dx.doi.org/10.1016/j.dam.2016.08.008 0166-218X/© 2016 Elsevier B.V. All rights reserved.
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Theorem 1.1 (Chen and Yesha [4]). A twin-free overlap-connected interval hypergraph without isolated vertices has a unique interval ordering (up to reversal). If we want to extend this result to CA hypergraphs, the property of being overlap-connected does obviously not suffice. For example, the twin-free hypergraph H = {a, b}, {a, b, c }, {b, c , d} is overlap-connected but has essentially different arc orderings. Hence, we need to assume a stronger kind of connectedness. When A and B are overlapping subsets of V (i.e., A G B) that additionally satisfy A ∪ B ̸= V , we say that A and B strictly overlap and write A G∗ B. Quilliot [19] proves that a CA hypergraph H on n vertices has a unique arc ordering if and only if for every set X ⊂ V (H ) with 1 < |X | < n − 1 there exists a hyperedge H ∈ H such that H G∗ X . Note that this criterion is not efficiently verifiable (at least not directly) as it involves quantification over exponentially many subsets X . We call a hypergraph H strictly overlap-connected if the graph (H , G∗ ) is connected. The treatment of interval hypergraphs in [4] can easily be adapted for proving a sufficient rigidity condition for CA hypergraphs: A twin-free, strictly overlapconnected CA hypergraph has a unique arc representation (up to reversal). Moreover, in Section 2 we prove the following criterion. Theorem 1.2. Given a CA hypergraph H on n ≥ 4 vertices, let H ′ be the hypergraph on the same vertex set obtained from H by removing all hyperedges of size 1, n − 1, and n. Then H has a unique arc ordering (up to reversal) if and only if H ′ is twin-free and strictly overlap-connected. 1.2. Tight orderings Let us denote by A ◃▹ B that two sets A and B have a non-empty intersection. By the standard terminology, a hypergraph
H is connected if the graph (H , ◃▹) is connected. Note that the assumption made in Theorem 1.1 cannot be weakened just to connectedness; consider H = {a}, {a, b, c } as the simplest example. Thus, if we want to weaken the assumption, we have also to weaken the conclusion. Call an arc ordering of a hypergraph H tight if, for any two hyperedges A and B such that A ⊆ B ̸= V , the corresponding arcs share an endpoint.2 The definition of a tight interval ordering is similar: We require that the intervals corresponding to hyperedges A and B share an endpoint whenever A ⊆ B (the condition B ̸= V is now dropped as the complete interval V has two endpoints, while the complete arc V has none). The class of hypergraphs admitting a tight interval ordering is characterized in terms of forbidden subhypergraphs in [17] (for interval hypergraphs, such a characterization is known due to [23]). Tight orderings inherently appear in the study of proper interval and proper circular-arc graphs; see the next subsection. Let A and B be nonempty sets. Note that A ◃▹ B iff A G B or A ⊆ B or A ⊇ B. Likewise, we define A ◃▹∗ B,
if A G∗ B or A ⊆ B or A ⊇ B,
and say that A and B strictly intersect. We call a hypergraph H strictly connected if the graph (H , ◃▹∗ ) is connected. In Section 2 we show that the approach of Chen and Yesha [4] works as well for tight orderings. Theorem 1.3. 1. A twin-free connected hypergraph without isolated vertices has at most one tight interval ordering (up to reversal). 2. A twin-free, strictly connected hypergraph has at most one tight arc ordering (up to reversal). 1.3. Neighborhood hypergraphs of PCA graphs For a vertex v of a graph G, the set of vertices adjacent to v is denoted by N (v). Furthermore, N [v] = N (v) ∪ {v}. We define the neighborhood hypergraph of G by N (G) = {N (v)}v∈V (G) and the closed neighborhood hypergraph of G by N [G] = {N [v]}v∈V (G) . An interval (resp. arc) representation of a graph G is a mapping from the vertex set of G to an interval (resp. CA) hypergraph H such that two vertices of G are adjacent exactly when the corresponding hyperedges of H have a nonempty intersection. Such a representation is proper if none of two hyperedges of H includes the other. Graphs having such representations are called proper interval and proper circular-arc (PCA) graphs. Roberts [20] discovered that G is a proper interval graph if and only if N [G] is an interval hypergraph. The case of PCA graphs is more complex. If G is a PCA graph, then N [G] is a CA hypergraph. The converse is not always true. The graphs whose closed neighborhood hypergraphs are circular-arc are known as concaveround graphs [1], and they contain PCA graphs as a proper subclass. Taking a closer look at the relationship between PCA graphs and CA hypergraphs, Tucker [24] distinguishes the case when the complement graph G is non-bipartite and shows that then G is PCA exactly when N [G] is CA. In general, G is a PCA graph if and only if the hypergraph N [G] has a tight arc ordering; cf. [15].
2 The endpoints of an arc A ̸= V are the two uniquely determined elements a− and a+ of V such that A is the chain with respect to the successor relation on V starting in a− and ending in a+ .
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Our second goal is to examine connectedness and rigidity properties of the neighborhood hypergraphs of PCA graphs. We call two vertices u and v of a graph G twins if N [u] = N [v]. Note that u and v are twins in the graph G if and only if they are twins in the hypergraph N [G]. Thus, the absence of twins in G is a necessary condition for the rigidity of N [G]. Another obvious necessary condition is the connectedness of G (and, hence, of N [G]).3 In some important cases, these conditions are also sufficient for being rigid, and Theorems 1.1 and 1.2 are an appropriate tool for showing this. Similarly, Theorem 1.3 is applicable if rigidity is considered with respect to tight orderings. Before implementing this scenario for PCA graphs, we give a brief overview of the classical case of proper interval graphs. Proper interval graphs. Let G be a proper interval graph. By Theorem 1.3.1 we know that N [G] has a unique tight interval ordering, if G is twin-free and connected. Under the same assumption, Roberts [20] proves that N [G] even has a unique interval ordering (implying that interval orderings of N [G] are tight). In [14] we noticed that the hypergraph N [G] \ {V (G)} is overlap-connected if G is connected. This reveals that Roberts’s rigidity result is a direct consequence of Theorem 1.1. Proper circular-arc graphs. Suppose now that G is a connected PCA graph with n vertices. Consider first the case that G is non-bipartite. In this case we prove two facts about connectedness properties of the hypergraph N [G]:
• N [G] is strictly connected (see Theorem 3.2); • removal of all (n − 1)-vertex hyperedges from N [G] results in a strictly overlap-connected hypergraph (see Theorem 3.4). If G also is twin-free, then the first of these results allows us to apply Theorem 1.3.2 and conclude that N [G] has a unique tight arc ordering. Since any arc ordering of N [G] proves to be tight, it is unique as well (see Theorem 3.3.1). This fact is actually well known, being a direct consequence of [12, Theorem 4.5]. More specifically, Huang [12] proves that a connected twin-free graph G has at most one, up to reversal, orientation as a local tournament if the complement graph G is nonbipartite or if it is connected.4 In the case that G is non-bipartite, any arc ordering of N [G] determines an orientation of G as a local transitive tournament and, moreover, one can show that distinct arc orderings determine distinct orientations. This readily implies the uniqueness of an arc ordering of N [G] stated above. Our approach to proving this fact is an interesting alternative to the theory of local tournament orientations as it reveals that the rigidity of N [G] can be derived solely from the connectedness properties of this hypergraph. If G is bipartite, it is convenient to switch to the complement hypergraph N [G] = {V (G) \ N [v]}v∈V (G) . This hypergraph can be shown to be interval. Hence, we can apply Theorem 1.3.1 to the connected components of N [G] and prove that if G is connected, then N [G] has exactly two tight arc orderings (up to reversal; see Theorem 3.3.2). It is curious to notice that, in contrast to Huang’s uniqueness result [12, Theorem 4.5], N [G] can in this case have exponentially many arc orderings (for an example see Section 3). The uniqueness result by Huang [12, Theorem 4.5] readily implies that any twin-free, connected PCA graph G with nonbipartite or connected complement has a unique proper arc representation; cf. [6, Corollary 2.9]. Speaking of uniqueness, we here suppose that an arc model of an n-vertex graph has 2n points and consists of arcs that do not share an endpoint. We conclude with noticing that also this important fact can alternatively be derived from the connectedness properties of the closed neighborhood hypergraphs of PCA graphs (see Theorem 3.5). 1.4. Relevance to isomorphism testing and canonization The rigidity results discussed above show that if a connected PCA graph G has a non-bipartite or a connected complement, then the adjacency relation of G induces a unique circular order on the vertices of G. Given G, such an order can be found very efficiently by using the algorithmic techniques designed in [2,10,11]. Since a circular order leaves at most 2n different possibilities to enumerate the vertices in a canonical way, this readily implies an efficient canonical labeling algorithm and, hence, an efficient isomorphism test for this class of graphs; see [16]. Moreover, the uniqueness of an intersection representation implies that the isomorphism problem reduces to the intersection representation problem (since two graphs are isomorphic if and only if their intersection models are congruent). Note further that the case of PCA graphs having a disconnected bipartite complement is not much harder and can be handled similarly. We refer the reader to [5,6,9,13,15,14,16,22] for algorithmic results in this area. Other algorithmic problems on the neighborhood hypergraphs of general graphs and their relevance to the graph isomorphism problem are discussed in [3]. 2. Interval and circular-arc hypergraphs Let V = {v1 , . . . , vn }. Saying that the sequence v1 , . . . , vn is circularly ordered, we mean that V is endowed with the circular successor relation ≺ under which vi ≺ vi+1 for i < n and vn ≺ v1 . An ordered pair of elements a− , a+ ∈ V determines an arc A = [a− , a+ ] that consists of the vertices appearing in the directed path from a− to a+ . This notation will
3 Small graphs are an exception, as all interval orderings of at most two vertices are the same up to reversal, and all arc orderings of up to three vertices are the same up to reversal. 4 In fact, if G is non-bipartite, then it is connected; cf. [12, Corollary 4.7].
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be used under the assumption that A ̸= V , though we also allow the complete arc A = V . By Cn we denote the set {1, . . . , n} endowed with the circular order 1 ≺ 2 ≺ · · · ≺ n ≺ 1. We will consider hypergraphs with nonempty hyperedges. An arc representation of a hypergraph H is a hypergraph isomorphism ρ from H to an arc system A on the circle Cn . The arc system A is referred to as an arc model of H . Note that H has an arc representation exactly when it admits an arc ordering ≺. Indeed, if ρ : V (H ) → {1, . . . , n} is an arc representation of H , we can define ≺ρ by ρ −1 (1) ≺ρ ρ −1 (2) ≺ρ . . . ≺ρ ρ −1 (n) ≺ρ ρ −1 (1). Conversely, if v1 ≺ v2 ≺ · · · ≺ vn ≺ v1 is an arc ordering of H , then ρ≺ (vi ) = i is an arc representation of H . Furthermore, we call ρ tight if the circular order of Cn is tight for A. Obviously, tight arc representations correspond to tight arc orderings and vice versa. The notion of an interval representation, corresponding to an interval ordering, is introduced similarly. The rotations x → (x + s) mod (n + 1) and the reflection x → n + 1 − x will be considered symmetries of the circle Cn . The linearly ordered segment {1, . . . , n} has a unique symmetry, namely the reflection. Note that, if ρ is an interval or arc representation of H and σ is a symmetry of the circle or the interval, respectively, then the composition σ ◦ ρ is an interval or arc representation of H as well. Turning back to the equivalence between arc representations and orderings, note that while ρ determines ≺ρ uniquely, ≺ determines ρ≺ up to rotation. Now, notice that H admits a unique, up to reversing, (tight) interval ordering if and only if H has a unique, up to reflection, (tight) interval representation. Furthermore, H admits a unique, up to reversing, (tight) arc ordering if and only if H has a unique, up to reflection and rotation, (tight) arc representation. Our proof strategy for the ‘‘if’’ part of Theorem 1.2 and for Theorem 1.3 is based on this correspondence. The following lemma translates these results into the language of interval/arc representations and, moreover, generalizes them to hypergraphs with twins. When applied to twin-free hypergraphs, this lemma implies the uniqueness of an interval/arc representation. Thus, the ‘‘if’’ part of Theorem 1.2 follows from Part 1 of the lemma, and Theorem 1.3 follows from Parts 4 and 3. Note also that Theorem 1.1 is a consequence of Part 2. Speaking of twin classes of a hypergraph, we mean the partition of its vertex set into equivalence classes under the equivalence relation of being twins. Lemma 2.1. 1. For any strictly overlap-connected CA hypergraph H , there is an enumeration T1 , . . . , Tm of its twin classes such that, for every arc representation ρ of H , each ρ(Tj ) is an arc, and the arcs ρ(T1 ), . . . , ρ(Tm ) appear on the circle in this order up to reflection and rotation. 2. For any overlap-connected interval hypergraph H with no isolated vertex, there is an enumeration T1 , . . . , Tm of its twin classes such that, for every interval representation ρ of H , each ρ(Tj ) is an interval, and the intervals ρ(T1 ), . . . , ρ(Tm ) appear in this or in the reverse order. 3. For any strictly connected CA hypergraph H , there is an enumeration T1 , . . . , Tm of its twin classes such that, for every tight arc representation ρ of H , each ρ(Tj ) is an arc, and the arcs ρ(T1 ), . . . , ρ(Tm ) appear on the circle in this order up to reflection and rotation. 4. For any connected interval hypergraph H with no isolated vertex, there is an enumeration T1 , . . . , Tm of its twin classes such that, for every tight interval representation ρ of H , each ρ(Tj ) is an interval, and the intervals ρ(T1 ), . . . , ρ(Tm ) appear in this or in the reverse order. Proof. The proof of each part follows the same scheme, used by Chen and Yesha [4, Theorem 2] for overlap-connected interval hypergraphs. Moreover, the claims about interval hypergraphs easily follow from the analogous claims about CA hypergraphs. 1. Let H1 , . . . , Hk be an enumeration of the hyperedges of H such that the hypergraph Hi = {H1 , . . . , Hi } is strictly overlap-connected for every i ≤ k. It is supposed that V (Hi ) = V (H ). We prove the claim for each Hi by induction on i (note that Hk = H ). For H1 , the claim is trivially true because this hypergraph has at most two twin classes. Assume that the claim is true for Hi , that is, there is an enumeration X1 , . . . , Xs of twin classes of this hypergraph such that, in every arc representation of Hi , they are arcs appearing on the circle in this order. Consider an arbitrary arc representation ρ of Hi+1 . Since ρ is an arc representation also for Hi , the induction assumption implies that the arc ρ(Hi+1 ) intersects a sequence of consecutive arcs ρ(Xa ), . . . , ρ(Xb ). Without loss of generality, we can suppose that 1 ≤ a ≤ b ≤ s. Thus, the hyperedge Hi+1 intersects exactly the twin classes Xa , . . . , Xb and, moreover, it includes the twin classes Xc for all c strictly between a and b. Note that neither Hi+1 nor its complement Hi+1 = V (H ) \ Hi+1 is included in any twin class Xj . Indeed, consider an arbitrary hyperedge Hl , l ≤ i, of Hi . If Hi+1 ⊆ Xj , then either Hl includes Xj and, hence, also Hi+1 , or Hl has an empty intersection with Xj and, hence, also with Hi+1 . If Hi+1 ⊆ Xj , then either Hl includes Xj and then Hl ∪ Hi+1 = V (H ), or Hl has an empty intersection with Xj and then Hl is contained in Hi+1 . In each case, the hypergraph Hi+1 could not be strictly overlap-connected. It follows from the above that a ̸= b. Adding Hi+1 to Hi can introduce new twin classes only by splitting Xa into Xa ∩ Hi+1 and Xa \ Hi+1 or by similarly splitting Xb . In the arc representation ρ , the arc ρ(Xa ) is split into two arcs ρ(Xa ∩ Hi+1 ) and ρ(Xa \ Hi+1 ), where the former adjoins the arc ρ(Xa+1 ). The similar holds true for ρ(Xb ). This implies that all twin classes of Hi+1 are arcs in ρ . Moreover, their circular order is predetermined by the sequence X1 , . . . , Xs and, therefore, does not depend on ρ .
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2. We derive this part from Part 1. Let H be an overlap-connected interval hypergraph with no isolated vertex. Let H ′ be obtained from H by adding an isolated vertex v . Note that H ′ is a strictly overlap-connected CA hypergraph. Note also that every twin class of H is a twin class also in H ′ . Denote the number of vertices in H by n. Any interval representation ρ : V (H ) → {1, . . . , n} of H can be extended to an arc representation ρ ′ of H ′ by defining ρ ′ (v) = n + 1 and regarding {1, . . . , n, n + 1} as the circle Cn+1 . If a twin class of H is not an interval in ρ , it is also not an arc in ρ ′ , giving a contradiction with Part 1. Furthermore, assume that H has interval representations ρ1 and ρ2 such that the orders of the twin classes of H in ρ1 and ρ2 are neither equal nor mutually reversed. Then the orders of the twin classes of H ′ in ρ1′ and ρ2′ are different, even up to reflection and rotation. This again contradicts Part 1. 3. First of all, we can assume that H has no complete hyperedge V (H ); otherwise it suffices to consider the hypergraph H \ {V (H )}, which has the same twin classes as H . We use an inductive argument as in Part 1. Enumerate the hyperedges of H = {H1 , . . . , Hk } so that all subhypergraphs Hi = {H1 , . . . , Hi } are strictly connected; remember that V (Hi ) = V (H ). In order to obtain the claim for H = Hk , we prove it for each Hi . The claim is obviously true for H1 and H2 . Let i ≥ 2 and assume that there is an enumeration X1 , . . . , Xs of twin classes of Hi such that they are arcs and appear in this circular order in every tight arc representation of Hi . Consider an arbitrary tight arc representation ρ of Hi+1 . By the induction assumption, the arc ρ(Hi+1 ) intersects a sequence of consecutive arcs ρ(Xa ), . . . , ρ(Xb ) and, therefore, the hyperedge Hi+1 intersects exactly the twin classes Xa , . . . , Xb . Moreover, Xc ⊆ Hi+1 whenever a < c < b. Thus, adding Hi+1 to Hi can only introduce new twin classes Xa ∩ Hi+1 and Xa \ Hi+1 or Xb ∩ Hi+1 and Xb \ Hi+1 . We will consider three cases. If neither Hi+1 nor Hi+1 is included in any twin class Xj , then, like in Part 1, the new twin classes are arcs in ρ whose position in the circle is predetermined by the sequence X1 , . . . , Xs and does not depend on ρ . Suppose now that Hi+1 ⊆ Xj for some j ≤ s. Unless Hi+1 = Xj , adding Hi+1 to Hi introduces two new twin classes Xj ∩ Hi+1 and Xj \ Hi+1 . The former twin class Xj must be contained in some hyperedge Hl of Hi for else Hi+1 could not be even connected. If the inclusion Xj ⊂ Hl is strict, then ρ(Xj ) and ρ(Hl ) must share an endpoint for else ρ(Hi+1 ) would be located strictly within ρ(Hl ) and ρ could not be tight. By the tightness of ρ , the arc ρ(Hi+1 ) must share an endpoint both with ρ(Xj ) and ρ(Hl ). Therefore, both ρ(Xj ∩ Hi+1 ) and ρ(Xj \ Hi+1 ) are arcs and their position in the circle does not depend on ρ , being predetermined by the sequence X1 , . . . , Xs . If Xj = Hl , then Xj must be strictly contained in some hyperedge Hl′ of Hi (because i ≥ 2), and the analysis is virtually the same. It remains to consider the case that Hi+1 ⊆ Xj for some j ≤ s. Suppose that this inclusion is strict; otherwise Hi+1 has the same twin classes as Hi , and we are done. Thus, adding Hi+1 to Hi introduces two new twin classes Xj ∩ Hi+1 and Xj \ Hi+1 . The former twin class Xj cannot be contained in all hyperedges Hl of Hi for else we would have Hi+1 ∪ Hl = V (H ) for all l ≤ i, and Hi+1 could not be strictly connected. Fix l such that Hl ∩ Xj = ∅. Since Hl ⊂ Hi+1 , the arcs ρ(Hl ) and ρ(Hi+1 ) must share an endpoint. The arc ρ(Hi+1 ) must contain at least one endpoint of the arc ρ(Xj ) but cannot contain both of them because then ρ(Hl ) would be strictly within ρ(Hi+1 ). This implies that both ρ(Xj ∩ Hi+1 ) and ρ(Xj \ Hi+1 ) are arcs. If Hl ̸= Xj , the position of these two arcs is determined by the condition that ρ(Xj \ Hi+1 ) adjoins ρ(Hl ) and, hence, is predetermined by the sequence X1 , . . . , Xs . If Hl = Xj , then Hl must properly include another hyperedge Hl′ of Hi (because i ≥ 2), and the position of the new arcs is determined by the contiguity to ρ(Hl′ ). 4. This part follows from Part 3 in the same way as Part 2 follows from Part 1. It remains to prove the ‘‘only if’’ part of Theorem 1.2. Lemma 2.2. Let H be a CA hypergraph on n ≥ 4 vertices and suppose that it has a unique arc ordering (up to reversal). Then the hypergraph H ′ on the same vertex set V = V (H ) obtained from H by removing all hyperedges of size 1, n − 1, and n is twin-free and strictly overlap-connected. Proof. Note that the hypergraphs H and H ′ have the same arc orderings. Thus, an arc ordering of H ′ is also unique up to reversal. Since n ≥ 4, H ′ has no twins for else transposition of two twins would give a different arc ordering. ′ Fix an arbitrary vertex x. For each hyperedge H ∈ H , set Hx = H if x ̸∈ H and Hx = V \ H otherwise. Consider the hypergraph Hx′ = Hx H ∈H ′ on the same vertex set V . The vertex x is isolated in Hx′ . Let Hx− denote the hypergraph obtained from Hx′ by removing x from it. Since a hyperedge is an arc with respect to a circular order on V exactly when its complement is an arc, the hypergraphs H ′ and Hx′ have the same arc orderings. Therefore, Hx′ has a unique arc ordering, and Hx− has a unique interval ordering. For two subsets A and B of V , note that A G∗ B exactly when A G∗ (V \ B). It follows that H ′ and Hx′ are simultaneously strictly overlap-connected or not. Furthermore, Hx′ is strictly overlap-connected if and only if Hx− is overlap-connected. This reduces our task to proving the overlap-connectedness of Hx− . Note that Hx− has no hyperedge of size 1 or n − 1. Therefore, it suffices to show that an m-vertex non-overlap-connected interval hypergraph K with no hyperedge of size 1 or m has at least two different interval orderings. By an overlap-connected component of K we will mean an inclusion-maximal overlap-connected subhypergraph K ′ ⊆ K . Note that two overlapconnected components either are vertex-disjoint or one is contained in a twin class of the other. Since K is not overlapconnected, it has more than one overlap-connected component. Consider one of them, K ′ , that does not contain any smaller
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component. Note that 1 < |V (K ′ )| < m. An interval ordering ≺ of K induces an interval ordering of K ′ . Reversing ≺ within V (K ′ ), we obtain a different ordering ≺′ on V (K ). It remains to note that ≺′ is an interval ordering of K because any hyperedge K ∈ K \ K ′ either is disjoint with V (K ′ ) or contains it. 3. The neighborhood hypergraphs of PCA graphs Let A be an arc system on the circle Cm . A bijection α : V (G) → A is an arc representation of a graph G if two vertices u and v are adjacent in G exactly when the arcs α(u) and α(v) intersect. A representation α is proper if α(u) ⊆ α(v) for no two vertices u and v . Restriction to intervals in the linearly ordered set {1, . . . , m} gives the notion of a proper interval representation of G. Graphs having such intersection representations are known as proper interval and proper circular-arc (PCA) graphs. A vertex u is called universal if it is adjacent to every other vertex, i.e., N [u] = V (G). The aim of this section is to examine the connectedness and rigidity properties of the closed neighborhood hypergraphs of PCA graphs. We begin with stating a known technical lemma. Any proper arc representation α : V (G) → A of a graph G determines a circular order on the vertex set V (G) accordingly to the appearance of the left (or, equivalently, right) endpoints of the arcs α(v), v ∈ V (G), in the circle Cm . We refer to this order as the geometric order associated with α and denote it by ≺α . Lemma 3.1. Let G be a PCA graph. 1. For any proper arc representation α of G, the associated geometric order ≺α on V (G) is a tight arc ordering of the hypergraph N [G]. 2. If the complement of G is not bipartite, then any arc ordering ≺ of N [G] is tight. Part 1 of Lemma 3.1 can be derived from the fact that, if no two arcs α(u) and α(v) cover the entire circle, then orienting each edge {u, v} as (u, v) if α(u) contains the left endpoint of α(v) results in a transitive local tournament [9,21]. A direct proof of this part can be found in [15]. Part 2 of Lemma 3.1 is a strengthening of Part 1 for non-co-bipartite PCA graphs. It says that all (not just geometric) arc orderings of N [G] are tight in this case. We thank the anonymous referee who noticed that this is a direct consequence of [1, Lemma 3.11]. The following fact shows that Theorem 1.3.2 applies to the closed neighborhood hypergraphs of connected non-cobipartite PCA graphs. Theorem 3.2. If G is a connected PCA graph with non-bipartite complement, then N [G] is strictly connected. Proof. Let α be a proper arc representation of G. By Lemma 3.1.1, ≺α is an arc ordering of N [G] (we here do not need the fact that ≺α is tight). Since G is connected, there is at most one pair of non-adjacent vertices x and y satisfying the relation x ≺α y. Therefore, all vertices of G can be arranged into a path v1 , . . . , vn such that vi and vi+1 are adjacent and vi ≺α vi+1 for every 1 ≤ i < n. We claim that N [vi ] ◃▹∗ N [vi+1 ] for each i < n. Indeed, note first that N [vi ] ∩ N [vi+1 ] ̸= ∅ as both sets contain the two vertices vi and vi+1 . It remains to prove that N [vi ] ∪ N [vi+1 ] ̸= V (G). Since G is a non-co-bipartite PCA graph, as it is well known, it does not contain universal vertices (otherwise, if N [u] = V (G), then every arc α(v) would contain one of the endpoints of the arc α(u), and this would determine a partition of V (G) into two cliques). Therefore, for any vertex u in G we can use notation N [u] = [u− , u+ ] with respect to ≺α , since the left endpoint u− and the right endpoint u+ are uniquely determined. Note that, for each u, both arcs [u− , u] and [u, u+ ] are cliques (because α(v) contains the left endpoint of α(u) if v ∈ [u− , u] and the right endpoint of α(u) if v ∈ [u, u+ ]). Note now that the equality N [vi ] ∪ N [vi+1 ] = V (G) would imply that [vi− , vi ] ∪ [vi+1 , vi++1 ] = V (G), yielding a partition of V (G) into two cliques. Thus, N [v1 ], . . . , N [vn ] is a strictly connected path passing through all hyperedges of N [G]. Now we are ready to prove the rigidity result for the closed neighborhood hypergraphs of PCA graphs. As discussed in Section 1, the present proof of Part 1 of the following theorem is alternative to the theory of local tournament orientations [12]. Theorem 3.3. Let G be a twin-free, connected PCA graph with more than one vertex. 1. If G is non-bipartite, then N [G] has a unique arc ordering (up to reversal). 2. If G is bipartite and connected, then N [G] has exactly two tight arc orderings (up to reversal). Proof. 1. The existence of an arc ordering of N [G] is a simple fact stated in Part 1 of Lemma 3.1. By Part 2 of Lemma 3.1, any arc ordering of N [G] is tight. The uniqueness follows from Theorem 1.3.2 by Theorem 3.2. 2. Recall that the open neighborhood hypergraph of a graph G is defined by N (G) = {N (v)}v∈V (G) . In place of N [G], it is now practical to consider the complement hypergraph N [G] = V (G) \ N [v] v∈V (G) . Note that N [G] = N (G). This hypergraph is disconnected, and any tight arc ordering of N [G] induces a tight interval ordering of each connected component of N (G). By Lemma 3.1.1, this applies to any geometric ordering of N [G]. Conversely, arbitrary tight interval orderings of the components of N (G) can be merged into a tight arc ordering of N [G]. Since G is connected, N (G) has exactly two components. Applying Theorem 1.3.1 to each of them, we conclude that each of the two components has a single tight interval ordering (up to reversal). Since there are exactly two essentially different ways to merge them, we see that N [G] has exactly two tight arc orderings (up to reversal).
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Fig. 1. An example: (a) A proper arc system; (b) The corresponding intersection graph G. Its complement G is non-bipartite; (c) The closed neighborhood hypergraph N [G] is not strictly overlap-connected: the hyperedge N [a] forms a single strictly overlap-connected component.
Fig. 2. A non-co-bipartite, twin-free, connected PCA graph on n vertices can have more than n/3 vertices of degree n − 2 (a vertex v has degree n − 2 iff |N [v]| = n − 1). For each k ≥ 2, define a graph Gk on 3k − 1 vertices by its arc model Ak such that k of the vertices will have degree n − 2. On the circle C = {u1 , u2 , . . . , uk , v1 , w1 , v2 , w2 , . . . , vk−1 , wk−1 , vk }, whose points go in this circular order, consider arcs U1 = [u1 , uk−1 ], Ui = [ui , vi−1 ] for 2 ≤ i ≤ k, V1 = [v1 , vk ], Vi = [vi , ui−1 ] = C \ Ui for 2 ≤ i ≤ k, and Wi = Vi \ {vi } for i ≤ k − 1. As Ak is tight, it can easily be made proper and, hence, Gk is PCA. Since the arcs have pairwise different left endpoints, we can identify V (Gk ) = C. The graph can be described by listing the pairs of non-adjacent vertices, namely vi ui , wi ui , wi ui+1 , and u1 uk . There are no twins, and Gk is not co-bipartite because its complement contains an odd cycle, namely u1 w1 u2 w2 . . . uk . In accordance with Theorem 3.4, when we remove from N [Gk ] the hyperedges N [vi ] = C \ {ui }, the hypergraph stays twinfree and becomes strictly overlap-connected. This can be seen by looking at the following path in the complemented hypergraph: N [u1 ] = {uk , v1 , w1 }, N [u2 ] = {w1 , v2 , w2 }, . . . , N [uk−1 ] = {wk−2 , vk−1 , wk−1 }, N [uk ] = {wk−1 , vk , u1 }, N [w1 ] = {u1 , u2 }, . . . , N [wk−1 ] = {uk−1 , uk }. The figure shows A3 and an arc model for N [G3 ] which has the arcs of size n − 1 grayed out.
Let us stress that Part 2 of Theorem 3.3 concerns only tight orderings. To show that it cannot be strengthened to the class of all arc orderings, consider the half-graph Hm that is the bipartite graph with vertex classes {u1 , . . . , um } and {v1 , . . . , vm } where ui is adjacent to vj if i ≤ j. The complement Gm = Hm is a twin-free, connected PCA graph. Note that, besides two pairs of mutually reversed tight interval orderings, the hypergraph N (H3 ) has another interval ordering and, hence, N [G3 ] has a non-tight arc ordering. If we increase the parameter m, the number of non-tight arc orderings of N [Gm ] grows exponentially. Let G be a twin-free, connected PCA graph with non-bipartite complement. In Theorem 3.2 we established that N [G] is strictly connected. We now take a closer look at the connectedness properties of this hypergraph. Note that N [G] is not always strictly overlap-connected: It can have hyperedges of size n − 1, and each such hyperedge forms a separate strictly overlap-connected component; see an example in Fig. 1. Nevertheless, if we remove the (n − 1)-element hyperedge from N [G] in this example, the remaining hypergraph becomes strictly overlap-connected and stays twin-free. It turns out that this is a general phenomenon. Theorem 3.4. If G is a twin-free, non-co-bipartite, connected PCA graph on n vertices, then the hypergraph obtained from N [G] by removing all hyperedges of size n − 1 is twin-free and strictly overlap-connected. Proof. Assume that a graph G satisfies all the assumptions. Note that then G has n ≥ 4 vertices. Denote H = N [G]. Since G is connected, H has no hyperedge of size 1. It also has no hyperedge of size n because G is non-co-bipartite and, hence, has no universal vertex. Remove all hyperedges of size n − 1 and denote the result by H ′ . By Theorem 3.3.1, H has a unique arc ordering (up to reversal). Therefore, H ′ is twin-free and strictly overlap-connected by Theorem 1.2. Note that Theorem 3.4 gives us a complete description of the decomposition of N [G] into strictly overlap-connected components because each hyperedge of size n − 1 forms such a component alone. Fig. 2 shows that the number of such single-hyperedge components can be linear.
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Fig. 3. Proof of Theorem 3.5.
We conclude with pointing out yet another application of the results of Section 2. Theorem 3.5 is a recasting of the well-known consequence of [12, Theorem 4.5] that any twin-free, connected PCA graph G with non-bipartite or connected complement has a unique proper arc representation. We show that Theorem 1.3 provides an alternative, rather short way of proving this important fact. Theorem 3.5. Let G be a twin-free, connected PCA graph. If its complement G is non-bipartite or connected, then all geometric orders of N [G] associated with proper arc representations of G are equal up to reversing. Proof. If G is non-bipartite, the result follows from Part 1 of Theorem 3.3. If G is bipartite and connected, Part 2 of Theorem 3.3 leaves two different possibilities. It turns out that, nevertheless, a geometric order is unique also in this case. Let V (G) = U ∪ W be the bipartition of G into two independent sets. Note that U and W span two connected components of the hypergraph N (G). Consider an arbitrary proper arc representation α of G and the corresponding geometric order ≺α on V (G). Like in the proof of Theorem 3.3.2, notice that ≺α is a tight interval order for each connected component of N (G) and, therefore, the restrictions of ≺α to U and W , each considered up to reversing, do not depend on α by Theorem 1.3.1. This still leaves two distinct possibilities of merging them into ≺α . We now show that one of them is actually ruled out, and this will prove that ≺α does not depend on α , if this order is considered up to reversing. Note that, since G is connected, G cannot have universal vertices. In this case it is well known and easy to see [13, Section 4.2] that any proper arc representation α of G has the following property: If v and v ′ are adjacent vertices of G, then the arcs α(v) and α(v ′ ) strictly overlap, that is, contain exactly one endpoint of each other. The assumptions of the theorem imply that both U and W contain at least two vertices. Let w and w ′ be two vertices in W . Since w and w ′ are not twins, they are distinguished by adjacency to some vertex u ∈ U. W.l.o.g. suppose that w and u are adjacent, while w ′ and u are not. Since w is not universal in G, there is a vertex u′ in U non-adjacent to w . Note that α(w ′ ) and α(u) contain different endpoints of α(w), and α(u′ ) and α(w) contain different endpoints of α(u); see Fig. 3. It follows that the arcs α(w ′ ), α(w), α(u), and α(u′ ) occur in the arc model exactly in this circular order, irrespective of whether α(w ′ ) and α(u′ ) intersect or not. Since the quadruple w ′ , w, u, u′ was chosen in terms of the graph G alone and α was supposed to be arbitrary, this conclusion holds true for any proper arc representation of G. Therefore, there is a unique way of merging the restrictions of ≺α to U and W so as to obtain an ordering of V (G) consistent with some geometric order. This proves the desired uniqueness result. Acknowledgments The first two authors are supported by DFG grant KO 1053/7–2. The third author is supported by DFG grant VE 652/1–1. We are grateful to the anonymous referee who provided us with numerous detailed comments and useful references to the earlier work on PCA graphs. We also thank another anonymous referee whose suggestion to extend the approach of Chen and Yesha [4, Theorem 2] to CA hypergraphs led to simplifying our original proof of Lemma 2.1. References [1] J. Bang-Jensen, J. Huang, A. Yeo, Convex-round and concave-round graphs, SIAM J. Discrete Math. 13 (2) (2000) 179–193. [2] K. Booth, G. Lueker, Testing for the consecutive ones property, interval graphs, and graph planarity using PQ -tree algorithms, J. Comput. System Sci. 13 (3) (1976) 335–379. [3] E. Boros, V. Gurvich, I.E. Zverovich, Neighborhood hypergraphs of bipartite graphs, J. Graph Theory 58 (1) (2008) 69–95. [4] L. Chen, Y. Yesha, Parallel recognition of the consecutive ones property with applications, J. Algorithms 12 (3) (1991) 375–392. [5] A.R. Curtis, M.C. Lin, R.M. McConnell, Y. Nussbaum, F.J. Soulignac, J.P. Spinrad, J.L. Szwarcfiter, Isomorphism of graph classes related to the circular-ones property, Discrete Math. Theor. Comput. Sci. 15 (1) (2013) 157–182. [6] X. Deng, P. Hell, J. Huang, Linear-time representation algorithms for proper circular-arc graphs and proper interval graphs, SIAM J. Comput. 25 (2) (1996) 390–403. [7] M. Dom, Algorithmic aspects of the consecutive-ones property, Bull. EATCS 98 (2009) 27–59. [8] F. Gavril, R.Y. Pinter, S. Zaks, Intersection representations of matrices by subtrees and unicycles on graphs, J. Discrete Algorithms 6 (2) (2008) 216–228. [9] P. Hell, J. Huang, Lexicographic orientation and representation algorithms for comparability graphs, proper circular arc graphs, and proper interval graphs, J. Graph Theory 20 (3) (1995) 361–374.
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W.-L. Hsu, A simple test for the consecutive ones property, J. Algorithms 43 (1) (2002) 1–16. W.-L. Hsu, R.M. McConnell, PC trees and circular-ones arrangements, Theoret. Comput. Sci. 296 (1) (2003) 99–116. 3. J. Huang, On the structure of local tournaments, J. Combin. Theory Ser. B 63 (1995) 200–221. H. Kaplan, Y. Nussbaum, Certifying algorithms for recognizing proper circular-arc graphs and unit circular-arc graphs, Discrete Appl. Math. 157 (15) (2009) 3216–3230. J. Köbler, S. Kuhnert, B. Laubner, O. Verbitsky, Interval graphs: Canonical representations in Logspace, SIAM J. Comput. 40 (5) (2011) 1292–1315. J. Köbler, S. Kuhnert, O. Verbitsky, Solving the canonical representation and star system problems for proper circular-arc graphs in logspace, J. Discrete Algorithms (2016) in press. http://dx.doi.org/10.1016/j.jda.2016.03.001. M.C. Lin, F.J. Soulignac, J.L. Szwarcfiter, A simple linear time algorithm for the isomorphism problem on proper circular-arc graphs, in: Proceedings of the 11th Scandinavian Workshop on Algorithm Theory, in: Lecture Notes in Computer Science, vol. 5124, Springer, 2008, pp. 355–366. J. Moore, Interval hypergraphs and D-interval hypergraphs, Discrete Math. 17 (1977) 173–179. A. Ouangraoua, A. Bergeron, K.M. Swenson, Theory and practice of ultra-perfection, J. Comput. Biol. 18 (9) (2011) 1219–1230. A. Quilliot, Circular representation problem on hypergraphs, Discrete Math. 51 (1984) 251–264. F. Roberts, On the compatibility between a graph and a simple order, J. Combin. Theory Ser. B 11 (1971) 28–38. D.J. Skrien, A relationship between triangulated graphs, comparability graphs, proper interval graphs, proper circular-arc graphs, and nested interval graphs, J. Graph Theory 6 (3) (1982) 309–316. F.J. Soulignac, Fully dynamic recognition of proper circular-arc graphs, Algorithmica 71 (4) (2015) 904–968. W. Trotter, J. Moore, Characterization problems for graphs, partially ordered sets, lattices, and families of sets, Discrete Math. 16 (1976) 361–381. A. Tucker, Matrix characterizations of circular-arc graphs, Pacific J. Math. 39 (1971) 535–545.
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Discrete Applied Mathematics journal homepage: www.elsevier.com/locate/dam
Circular-arc hypergraphs: Rigidity via connectedness Johannes Köbler, Sebastian Kuhnert, Oleg Verbitsky ∗,1 Humboldt-Universität zu Berlin, Institut für Informatik, Unter den Linden 6, 10099 Berlin, Germany
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Article history: Received 18 September 2015 Received in revised form 1 June 2016 Accepted 16 August 2016 Available online xxxx Keywords: Circular-arc hypergraphs Circular-ones property Intersection graphs Proper circular-arc graphs Unique representations Graph canonization
abstract A circular-arc hypergraph H is a hypergraph admitting an arc ordering, that is, a circular ordering of the vertex set V (H ) such that every hyperedge is an arc of consecutive vertices. We give a criterion for the uniqueness of an arc ordering in terms of connectedness properties of H . This generalizes the relationship between rigidity and connectedness disclosed by Chen and Yesha (1991) in the case of interval hypergraphs. Moreover, we state sufficient conditions for the uniqueness of tight arc orderings where, for any two hyperedges A and B such that A ⊆ B ̸= V (H ), the corresponding arcs must share a common endpoint. We notice that these conditions are obeyed for the closed neighborhood hypergraphs of proper circular-arc graphs, implying for them the known rigidity results that were originally obtained using the theory of local tournament graph orientations. © 2016 Elsevier B.V. All rights reserved.
1. Introduction 1.1. Interval and circular-arc hypergraphs An interval ordering of a hypergraph H with a finite vertex set V = V (H ) is a linear ordering v1 , . . . , vn of V such that every hyperedge of H is an interval of consecutive vertices. This notion can be generalized to an arc ordering v1 , . . . , vn where the vertices are circularly ordered (i.e., v1 succeeds vn ) so that every hyperedge is an arc of consecutive vertices. An interval hypergraph is a hypergraph admitting an interval ordering. Similarly, if a hypergraph admits an arc ordering, we call it circular-arc (using also the shorthand CA). In the terminology stemming from computational genomics, interval hypergraphs are exactly those hypergraphs whose incidence matrix has the consecutive ones property; see, e.g., [7]. Similarly, a hypergraph is CA exactly when its incidence matrix has the circular ones property; see [8,18] for the relevance to computational genomics and [10,11] for the algorithmic aspects. Our goal is to study the conditions under which interval and circular-arc hypergraphs are rigid in the sense that they have a unique interval or arc ordering, respectively. Since any interval (or arc) ordering can be changed to another interval (or arc) ordering by reversing, we always mean uniqueness up to reversal. An obvious necessary condition for being rigid is that a hypergraph has no twins, that is, no two vertices such that every hyperedge contains either both or none of them. We say that two sets A and B overlap and write A G B, if A and B have nonempty intersection and neither of the two sets includes the other. To facilitate notation, we use the same character H to denote a hypergraph and the set of its hyperedges. We call H overlap-connected if the graph (H , G) is connected. A vertex of H is isolated if it is not contained in any hyperedge. As a starting point, we refer to the following rigidity result.
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Corresponding author. E-mail address: [email protected] (O. Verbitsky).
1 On leave from the Institute for Applied Problems of Mechanics and Mathematics, Lviv, Ukraine. http://dx.doi.org/10.1016/j.dam.2016.08.008 0166-218X/© 2016 Elsevier B.V. All rights reserved.
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Theorem 1.1 (Chen and Yesha [4]). A twin-free overlap-connected interval hypergraph without isolated vertices has a unique interval ordering (up to reversal). If we want to extend this result to CA hypergraphs, the property of being overlap-connected does obviously not suffice. For example, the twin-free hypergraph H = {a, b}, {a, b, c }, {b, c , d} is overlap-connected but has essentially different arc orderings. Hence, we need to assume a stronger kind of connectedness. When A and B are overlapping subsets of V (i.e., A G B) that additionally satisfy A ∪ B ̸= V , we say that A and B strictly overlap and write A G∗ B. Quilliot [19] proves that a CA hypergraph H on n vertices has a unique arc ordering if and only if for every set X ⊂ V (H ) with 1 < |X | < n − 1 there exists a hyperedge H ∈ H such that H G∗ X . Note that this criterion is not efficiently verifiable (at least not directly) as it involves quantification over exponentially many subsets X . We call a hypergraph H strictly overlap-connected if the graph (H , G∗ ) is connected. The treatment of interval hypergraphs in [4] can easily be adapted for proving a sufficient rigidity condition for CA hypergraphs: A twin-free, strictly overlapconnected CA hypergraph has a unique arc representation (up to reversal). Moreover, in Section 2 we prove the following criterion. Theorem 1.2. Given a CA hypergraph H on n ≥ 4 vertices, let H ′ be the hypergraph on the same vertex set obtained from H by removing all hyperedges of size 1, n − 1, and n. Then H has a unique arc ordering (up to reversal) if and only if H ′ is twin-free and strictly overlap-connected. 1.2. Tight orderings Let us denote by A ◃▹ B that two sets A and B have a non-empty intersection. By the standard terminology, a hypergraph
H is connected if the graph (H , ◃▹) is connected. Note that the assumption made in Theorem 1.1 cannot be weakened just to connectedness; consider H = {a}, {a, b, c } as the simplest example. Thus, if we want to weaken the assumption, we have also to weaken the conclusion. Call an arc ordering of a hypergraph H tight if, for any two hyperedges A and B such that A ⊆ B ̸= V , the corresponding arcs share an endpoint.2 The definition of a tight interval ordering is similar: We require that the intervals corresponding to hyperedges A and B share an endpoint whenever A ⊆ B (the condition B ̸= V is now dropped as the complete interval V has two endpoints, while the complete arc V has none). The class of hypergraphs admitting a tight interval ordering is characterized in terms of forbidden subhypergraphs in [17] (for interval hypergraphs, such a characterization is known due to [23]). Tight orderings inherently appear in the study of proper interval and proper circular-arc graphs; see the next subsection. Let A and B be nonempty sets. Note that A ◃▹ B iff A G B or A ⊆ B or A ⊇ B. Likewise, we define A ◃▹∗ B,
if A G∗ B or A ⊆ B or A ⊇ B,
and say that A and B strictly intersect. We call a hypergraph H strictly connected if the graph (H , ◃▹∗ ) is connected. In Section 2 we show that the approach of Chen and Yesha [4] works as well for tight orderings. Theorem 1.3. 1. A twin-free connected hypergraph without isolated vertices has at most one tight interval ordering (up to reversal). 2. A twin-free, strictly connected hypergraph has at most one tight arc ordering (up to reversal). 1.3. Neighborhood hypergraphs of PCA graphs For a vertex v of a graph G, the set of vertices adjacent to v is denoted by N (v). Furthermore, N [v] = N (v) ∪ {v}. We define the neighborhood hypergraph of G by N (G) = {N (v)}v∈V (G) and the closed neighborhood hypergraph of G by N [G] = {N [v]}v∈V (G) . An interval (resp. arc) representation of a graph G is a mapping from the vertex set of G to an interval (resp. CA) hypergraph H such that two vertices of G are adjacent exactly when the corresponding hyperedges of H have a nonempty intersection. Such a representation is proper if none of two hyperedges of H includes the other. Graphs having such representations are called proper interval and proper circular-arc (PCA) graphs. Roberts [20] discovered that G is a proper interval graph if and only if N [G] is an interval hypergraph. The case of PCA graphs is more complex. If G is a PCA graph, then N [G] is a CA hypergraph. The converse is not always true. The graphs whose closed neighborhood hypergraphs are circular-arc are known as concaveround graphs [1], and they contain PCA graphs as a proper subclass. Taking a closer look at the relationship between PCA graphs and CA hypergraphs, Tucker [24] distinguishes the case when the complement graph G is non-bipartite and shows that then G is PCA exactly when N [G] is CA. In general, G is a PCA graph if and only if the hypergraph N [G] has a tight arc ordering; cf. [15].
2 The endpoints of an arc A ̸= V are the two uniquely determined elements a− and a+ of V such that A is the chain with respect to the successor relation on V starting in a− and ending in a+ .
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Our second goal is to examine connectedness and rigidity properties of the neighborhood hypergraphs of PCA graphs. We call two vertices u and v of a graph G twins if N [u] = N [v]. Note that u and v are twins in the graph G if and only if they are twins in the hypergraph N [G]. Thus, the absence of twins in G is a necessary condition for the rigidity of N [G]. Another obvious necessary condition is the connectedness of G (and, hence, of N [G]).3 In some important cases, these conditions are also sufficient for being rigid, and Theorems 1.1 and 1.2 are an appropriate tool for showing this. Similarly, Theorem 1.3 is applicable if rigidity is considered with respect to tight orderings. Before implementing this scenario for PCA graphs, we give a brief overview of the classical case of proper interval graphs. Proper interval graphs. Let G be a proper interval graph. By Theorem 1.3.1 we know that N [G] has a unique tight interval ordering, if G is twin-free and connected. Under the same assumption, Roberts [20] proves that N [G] even has a unique interval ordering (implying that interval orderings of N [G] are tight). In [14] we noticed that the hypergraph N [G] \ {V (G)} is overlap-connected if G is connected. This reveals that Roberts’s rigidity result is a direct consequence of Theorem 1.1. Proper circular-arc graphs. Suppose now that G is a connected PCA graph with n vertices. Consider first the case that G is non-bipartite. In this case we prove two facts about connectedness properties of the hypergraph N [G]:
• N [G] is strictly connected (see Theorem 3.2); • removal of all (n − 1)-vertex hyperedges from N [G] results in a strictly overlap-connected hypergraph (see Theorem 3.4). If G also is twin-free, then the first of these results allows us to apply Theorem 1.3.2 and conclude that N [G] has a unique tight arc ordering. Since any arc ordering of N [G] proves to be tight, it is unique as well (see Theorem 3.3.1). This fact is actually well known, being a direct consequence of [12, Theorem 4.5]. More specifically, Huang [12] proves that a connected twin-free graph G has at most one, up to reversal, orientation as a local tournament if the complement graph G is nonbipartite or if it is connected.4 In the case that G is non-bipartite, any arc ordering of N [G] determines an orientation of G as a local transitive tournament and, moreover, one can show that distinct arc orderings determine distinct orientations. This readily implies the uniqueness of an arc ordering of N [G] stated above. Our approach to proving this fact is an interesting alternative to the theory of local tournament orientations as it reveals that the rigidity of N [G] can be derived solely from the connectedness properties of this hypergraph. If G is bipartite, it is convenient to switch to the complement hypergraph N [G] = {V (G) \ N [v]}v∈V (G) . This hypergraph can be shown to be interval. Hence, we can apply Theorem 1.3.1 to the connected components of N [G] and prove that if G is connected, then N [G] has exactly two tight arc orderings (up to reversal; see Theorem 3.3.2). It is curious to notice that, in contrast to Huang’s uniqueness result [12, Theorem 4.5], N [G] can in this case have exponentially many arc orderings (for an example see Section 3). The uniqueness result by Huang [12, Theorem 4.5] readily implies that any twin-free, connected PCA graph G with nonbipartite or connected complement has a unique proper arc representation; cf. [6, Corollary 2.9]. Speaking of uniqueness, we here suppose that an arc model of an n-vertex graph has 2n points and consists of arcs that do not share an endpoint. We conclude with noticing that also this important fact can alternatively be derived from the connectedness properties of the closed neighborhood hypergraphs of PCA graphs (see Theorem 3.5). 1.4. Relevance to isomorphism testing and canonization The rigidity results discussed above show that if a connected PCA graph G has a non-bipartite or a connected complement, then the adjacency relation of G induces a unique circular order on the vertices of G. Given G, such an order can be found very efficiently by using the algorithmic techniques designed in [2,10,11]. Since a circular order leaves at most 2n different possibilities to enumerate the vertices in a canonical way, this readily implies an efficient canonical labeling algorithm and, hence, an efficient isomorphism test for this class of graphs; see [16]. Moreover, the uniqueness of an intersection representation implies that the isomorphism problem reduces to the intersection representation problem (since two graphs are isomorphic if and only if their intersection models are congruent). Note further that the case of PCA graphs having a disconnected bipartite complement is not much harder and can be handled similarly. We refer the reader to [5,6,9,13,15,14,16,22] for algorithmic results in this area. Other algorithmic problems on the neighborhood hypergraphs of general graphs and their relevance to the graph isomorphism problem are discussed in [3]. 2. Interval and circular-arc hypergraphs Let V = {v1 , . . . , vn }. Saying that the sequence v1 , . . . , vn is circularly ordered, we mean that V is endowed with the circular successor relation ≺ under which vi ≺ vi+1 for i < n and vn ≺ v1 . An ordered pair of elements a− , a+ ∈ V determines an arc A = [a− , a+ ] that consists of the vertices appearing in the directed path from a− to a+ . This notation will
3 Small graphs are an exception, as all interval orderings of at most two vertices are the same up to reversal, and all arc orderings of up to three vertices are the same up to reversal. 4 In fact, if G is non-bipartite, then it is connected; cf. [12, Corollary 4.7].
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be used under the assumption that A ̸= V , though we also allow the complete arc A = V . By Cn we denote the set {1, . . . , n} endowed with the circular order 1 ≺ 2 ≺ · · · ≺ n ≺ 1. We will consider hypergraphs with nonempty hyperedges. An arc representation of a hypergraph H is a hypergraph isomorphism ρ from H to an arc system A on the circle Cn . The arc system A is referred to as an arc model of H . Note that H has an arc representation exactly when it admits an arc ordering ≺. Indeed, if ρ : V (H ) → {1, . . . , n} is an arc representation of H , we can define ≺ρ by ρ −1 (1) ≺ρ ρ −1 (2) ≺ρ . . . ≺ρ ρ −1 (n) ≺ρ ρ −1 (1). Conversely, if v1 ≺ v2 ≺ · · · ≺ vn ≺ v1 is an arc ordering of H , then ρ≺ (vi ) = i is an arc representation of H . Furthermore, we call ρ tight if the circular order of Cn is tight for A. Obviously, tight arc representations correspond to tight arc orderings and vice versa. The notion of an interval representation, corresponding to an interval ordering, is introduced similarly. The rotations x → (x + s) mod (n + 1) and the reflection x → n + 1 − x will be considered symmetries of the circle Cn . The linearly ordered segment {1, . . . , n} has a unique symmetry, namely the reflection. Note that, if ρ is an interval or arc representation of H and σ is a symmetry of the circle or the interval, respectively, then the composition σ ◦ ρ is an interval or arc representation of H as well. Turning back to the equivalence between arc representations and orderings, note that while ρ determines ≺ρ uniquely, ≺ determines ρ≺ up to rotation. Now, notice that H admits a unique, up to reversing, (tight) interval ordering if and only if H has a unique, up to reflection, (tight) interval representation. Furthermore, H admits a unique, up to reversing, (tight) arc ordering if and only if H has a unique, up to reflection and rotation, (tight) arc representation. Our proof strategy for the ‘‘if’’ part of Theorem 1.2 and for Theorem 1.3 is based on this correspondence. The following lemma translates these results into the language of interval/arc representations and, moreover, generalizes them to hypergraphs with twins. When applied to twin-free hypergraphs, this lemma implies the uniqueness of an interval/arc representation. Thus, the ‘‘if’’ part of Theorem 1.2 follows from Part 1 of the lemma, and Theorem 1.3 follows from Parts 4 and 3. Note also that Theorem 1.1 is a consequence of Part 2. Speaking of twin classes of a hypergraph, we mean the partition of its vertex set into equivalence classes under the equivalence relation of being twins. Lemma 2.1. 1. For any strictly overlap-connected CA hypergraph H , there is an enumeration T1 , . . . , Tm of its twin classes such that, for every arc representation ρ of H , each ρ(Tj ) is an arc, and the arcs ρ(T1 ), . . . , ρ(Tm ) appear on the circle in this order up to reflection and rotation. 2. For any overlap-connected interval hypergraph H with no isolated vertex, there is an enumeration T1 , . . . , Tm of its twin classes such that, for every interval representation ρ of H , each ρ(Tj ) is an interval, and the intervals ρ(T1 ), . . . , ρ(Tm ) appear in this or in the reverse order. 3. For any strictly connected CA hypergraph H , there is an enumeration T1 , . . . , Tm of its twin classes such that, for every tight arc representation ρ of H , each ρ(Tj ) is an arc, and the arcs ρ(T1 ), . . . , ρ(Tm ) appear on the circle in this order up to reflection and rotation. 4. For any connected interval hypergraph H with no isolated vertex, there is an enumeration T1 , . . . , Tm of its twin classes such that, for every tight interval representation ρ of H , each ρ(Tj ) is an interval, and the intervals ρ(T1 ), . . . , ρ(Tm ) appear in this or in the reverse order. Proof. The proof of each part follows the same scheme, used by Chen and Yesha [4, Theorem 2] for overlap-connected interval hypergraphs. Moreover, the claims about interval hypergraphs easily follow from the analogous claims about CA hypergraphs. 1. Let H1 , . . . , Hk be an enumeration of the hyperedges of H such that the hypergraph Hi = {H1 , . . . , Hi } is strictly overlap-connected for every i ≤ k. It is supposed that V (Hi ) = V (H ). We prove the claim for each Hi by induction on i (note that Hk = H ). For H1 , the claim is trivially true because this hypergraph has at most two twin classes. Assume that the claim is true for Hi , that is, there is an enumeration X1 , . . . , Xs of twin classes of this hypergraph such that, in every arc representation of Hi , they are arcs appearing on the circle in this order. Consider an arbitrary arc representation ρ of Hi+1 . Since ρ is an arc representation also for Hi , the induction assumption implies that the arc ρ(Hi+1 ) intersects a sequence of consecutive arcs ρ(Xa ), . . . , ρ(Xb ). Without loss of generality, we can suppose that 1 ≤ a ≤ b ≤ s. Thus, the hyperedge Hi+1 intersects exactly the twin classes Xa , . . . , Xb and, moreover, it includes the twin classes Xc for all c strictly between a and b. Note that neither Hi+1 nor its complement Hi+1 = V (H ) \ Hi+1 is included in any twin class Xj . Indeed, consider an arbitrary hyperedge Hl , l ≤ i, of Hi . If Hi+1 ⊆ Xj , then either Hl includes Xj and, hence, also Hi+1 , or Hl has an empty intersection with Xj and, hence, also with Hi+1 . If Hi+1 ⊆ Xj , then either Hl includes Xj and then Hl ∪ Hi+1 = V (H ), or Hl has an empty intersection with Xj and then Hl is contained in Hi+1 . In each case, the hypergraph Hi+1 could not be strictly overlap-connected. It follows from the above that a ̸= b. Adding Hi+1 to Hi can introduce new twin classes only by splitting Xa into Xa ∩ Hi+1 and Xa \ Hi+1 or by similarly splitting Xb . In the arc representation ρ , the arc ρ(Xa ) is split into two arcs ρ(Xa ∩ Hi+1 ) and ρ(Xa \ Hi+1 ), where the former adjoins the arc ρ(Xa+1 ). The similar holds true for ρ(Xb ). This implies that all twin classes of Hi+1 are arcs in ρ . Moreover, their circular order is predetermined by the sequence X1 , . . . , Xs and, therefore, does not depend on ρ .
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2. We derive this part from Part 1. Let H be an overlap-connected interval hypergraph with no isolated vertex. Let H ′ be obtained from H by adding an isolated vertex v . Note that H ′ is a strictly overlap-connected CA hypergraph. Note also that every twin class of H is a twin class also in H ′ . Denote the number of vertices in H by n. Any interval representation ρ : V (H ) → {1, . . . , n} of H can be extended to an arc representation ρ ′ of H ′ by defining ρ ′ (v) = n + 1 and regarding {1, . . . , n, n + 1} as the circle Cn+1 . If a twin class of H is not an interval in ρ , it is also not an arc in ρ ′ , giving a contradiction with Part 1. Furthermore, assume that H has interval representations ρ1 and ρ2 such that the orders of the twin classes of H in ρ1 and ρ2 are neither equal nor mutually reversed. Then the orders of the twin classes of H ′ in ρ1′ and ρ2′ are different, even up to reflection and rotation. This again contradicts Part 1. 3. First of all, we can assume that H has no complete hyperedge V (H ); otherwise it suffices to consider the hypergraph H \ {V (H )}, which has the same twin classes as H . We use an inductive argument as in Part 1. Enumerate the hyperedges of H = {H1 , . . . , Hk } so that all subhypergraphs Hi = {H1 , . . . , Hi } are strictly connected; remember that V (Hi ) = V (H ). In order to obtain the claim for H = Hk , we prove it for each Hi . The claim is obviously true for H1 and H2 . Let i ≥ 2 and assume that there is an enumeration X1 , . . . , Xs of twin classes of Hi such that they are arcs and appear in this circular order in every tight arc representation of Hi . Consider an arbitrary tight arc representation ρ of Hi+1 . By the induction assumption, the arc ρ(Hi+1 ) intersects a sequence of consecutive arcs ρ(Xa ), . . . , ρ(Xb ) and, therefore, the hyperedge Hi+1 intersects exactly the twin classes Xa , . . . , Xb . Moreover, Xc ⊆ Hi+1 whenever a < c < b. Thus, adding Hi+1 to Hi can only introduce new twin classes Xa ∩ Hi+1 and Xa \ Hi+1 or Xb ∩ Hi+1 and Xb \ Hi+1 . We will consider three cases. If neither Hi+1 nor Hi+1 is included in any twin class Xj , then, like in Part 1, the new twin classes are arcs in ρ whose position in the circle is predetermined by the sequence X1 , . . . , Xs and does not depend on ρ . Suppose now that Hi+1 ⊆ Xj for some j ≤ s. Unless Hi+1 = Xj , adding Hi+1 to Hi introduces two new twin classes Xj ∩ Hi+1 and Xj \ Hi+1 . The former twin class Xj must be contained in some hyperedge Hl of Hi for else Hi+1 could not be even connected. If the inclusion Xj ⊂ Hl is strict, then ρ(Xj ) and ρ(Hl ) must share an endpoint for else ρ(Hi+1 ) would be located strictly within ρ(Hl ) and ρ could not be tight. By the tightness of ρ , the arc ρ(Hi+1 ) must share an endpoint both with ρ(Xj ) and ρ(Hl ). Therefore, both ρ(Xj ∩ Hi+1 ) and ρ(Xj \ Hi+1 ) are arcs and their position in the circle does not depend on ρ , being predetermined by the sequence X1 , . . . , Xs . If Xj = Hl , then Xj must be strictly contained in some hyperedge Hl′ of Hi (because i ≥ 2), and the analysis is virtually the same. It remains to consider the case that Hi+1 ⊆ Xj for some j ≤ s. Suppose that this inclusion is strict; otherwise Hi+1 has the same twin classes as Hi , and we are done. Thus, adding Hi+1 to Hi introduces two new twin classes Xj ∩ Hi+1 and Xj \ Hi+1 . The former twin class Xj cannot be contained in all hyperedges Hl of Hi for else we would have Hi+1 ∪ Hl = V (H ) for all l ≤ i, and Hi+1 could not be strictly connected. Fix l such that Hl ∩ Xj = ∅. Since Hl ⊂ Hi+1 , the arcs ρ(Hl ) and ρ(Hi+1 ) must share an endpoint. The arc ρ(Hi+1 ) must contain at least one endpoint of the arc ρ(Xj ) but cannot contain both of them because then ρ(Hl ) would be strictly within ρ(Hi+1 ). This implies that both ρ(Xj ∩ Hi+1 ) and ρ(Xj \ Hi+1 ) are arcs. If Hl ̸= Xj , the position of these two arcs is determined by the condition that ρ(Xj \ Hi+1 ) adjoins ρ(Hl ) and, hence, is predetermined by the sequence X1 , . . . , Xs . If Hl = Xj , then Hl must properly include another hyperedge Hl′ of Hi (because i ≥ 2), and the position of the new arcs is determined by the contiguity to ρ(Hl′ ). 4. This part follows from Part 3 in the same way as Part 2 follows from Part 1. It remains to prove the ‘‘only if’’ part of Theorem 1.2. Lemma 2.2. Let H be a CA hypergraph on n ≥ 4 vertices and suppose that it has a unique arc ordering (up to reversal). Then the hypergraph H ′ on the same vertex set V = V (H ) obtained from H by removing all hyperedges of size 1, n − 1, and n is twin-free and strictly overlap-connected. Proof. Note that the hypergraphs H and H ′ have the same arc orderings. Thus, an arc ordering of H ′ is also unique up to reversal. Since n ≥ 4, H ′ has no twins for else transposition of two twins would give a different arc ordering. ′ Fix an arbitrary vertex x. For each hyperedge H ∈ H , set Hx = H if x ̸∈ H and Hx = V \ H otherwise. Consider the hypergraph Hx′ = Hx H ∈H ′ on the same vertex set V . The vertex x is isolated in Hx′ . Let Hx− denote the hypergraph obtained from Hx′ by removing x from it. Since a hyperedge is an arc with respect to a circular order on V exactly when its complement is an arc, the hypergraphs H ′ and Hx′ have the same arc orderings. Therefore, Hx′ has a unique arc ordering, and Hx− has a unique interval ordering. For two subsets A and B of V , note that A G∗ B exactly when A G∗ (V \ B). It follows that H ′ and Hx′ are simultaneously strictly overlap-connected or not. Furthermore, Hx′ is strictly overlap-connected if and only if Hx− is overlap-connected. This reduces our task to proving the overlap-connectedness of Hx− . Note that Hx− has no hyperedge of size 1 or n − 1. Therefore, it suffices to show that an m-vertex non-overlap-connected interval hypergraph K with no hyperedge of size 1 or m has at least two different interval orderings. By an overlap-connected component of K we will mean an inclusion-maximal overlap-connected subhypergraph K ′ ⊆ K . Note that two overlapconnected components either are vertex-disjoint or one is contained in a twin class of the other. Since K is not overlapconnected, it has more than one overlap-connected component. Consider one of them, K ′ , that does not contain any smaller
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component. Note that 1 < |V (K ′ )| < m. An interval ordering ≺ of K induces an interval ordering of K ′ . Reversing ≺ within V (K ′ ), we obtain a different ordering ≺′ on V (K ). It remains to note that ≺′ is an interval ordering of K because any hyperedge K ∈ K \ K ′ either is disjoint with V (K ′ ) or contains it. 3. The neighborhood hypergraphs of PCA graphs Let A be an arc system on the circle Cm . A bijection α : V (G) → A is an arc representation of a graph G if two vertices u and v are adjacent in G exactly when the arcs α(u) and α(v) intersect. A representation α is proper if α(u) ⊆ α(v) for no two vertices u and v . Restriction to intervals in the linearly ordered set {1, . . . , m} gives the notion of a proper interval representation of G. Graphs having such intersection representations are known as proper interval and proper circular-arc (PCA) graphs. A vertex u is called universal if it is adjacent to every other vertex, i.e., N [u] = V (G). The aim of this section is to examine the connectedness and rigidity properties of the closed neighborhood hypergraphs of PCA graphs. We begin with stating a known technical lemma. Any proper arc representation α : V (G) → A of a graph G determines a circular order on the vertex set V (G) accordingly to the appearance of the left (or, equivalently, right) endpoints of the arcs α(v), v ∈ V (G), in the circle Cm . We refer to this order as the geometric order associated with α and denote it by ≺α . Lemma 3.1. Let G be a PCA graph. 1. For any proper arc representation α of G, the associated geometric order ≺α on V (G) is a tight arc ordering of the hypergraph N [G]. 2. If the complement of G is not bipartite, then any arc ordering ≺ of N [G] is tight. Part 1 of Lemma 3.1 can be derived from the fact that, if no two arcs α(u) and α(v) cover the entire circle, then orienting each edge {u, v} as (u, v) if α(u) contains the left endpoint of α(v) results in a transitive local tournament [9,21]. A direct proof of this part can be found in [15]. Part 2 of Lemma 3.1 is a strengthening of Part 1 for non-co-bipartite PCA graphs. It says that all (not just geometric) arc orderings of N [G] are tight in this case. We thank the anonymous referee who noticed that this is a direct consequence of [1, Lemma 3.11]. The following fact shows that Theorem 1.3.2 applies to the closed neighborhood hypergraphs of connected non-cobipartite PCA graphs. Theorem 3.2. If G is a connected PCA graph with non-bipartite complement, then N [G] is strictly connected. Proof. Let α be a proper arc representation of G. By Lemma 3.1.1, ≺α is an arc ordering of N [G] (we here do not need the fact that ≺α is tight). Since G is connected, there is at most one pair of non-adjacent vertices x and y satisfying the relation x ≺α y. Therefore, all vertices of G can be arranged into a path v1 , . . . , vn such that vi and vi+1 are adjacent and vi ≺α vi+1 for every 1 ≤ i < n. We claim that N [vi ] ◃▹∗ N [vi+1 ] for each i < n. Indeed, note first that N [vi ] ∩ N [vi+1 ] ̸= ∅ as both sets contain the two vertices vi and vi+1 . It remains to prove that N [vi ] ∪ N [vi+1 ] ̸= V (G). Since G is a non-co-bipartite PCA graph, as it is well known, it does not contain universal vertices (otherwise, if N [u] = V (G), then every arc α(v) would contain one of the endpoints of the arc α(u), and this would determine a partition of V (G) into two cliques). Therefore, for any vertex u in G we can use notation N [u] = [u− , u+ ] with respect to ≺α , since the left endpoint u− and the right endpoint u+ are uniquely determined. Note that, for each u, both arcs [u− , u] and [u, u+ ] are cliques (because α(v) contains the left endpoint of α(u) if v ∈ [u− , u] and the right endpoint of α(u) if v ∈ [u, u+ ]). Note now that the equality N [vi ] ∪ N [vi+1 ] = V (G) would imply that [vi− , vi ] ∪ [vi+1 , vi++1 ] = V (G), yielding a partition of V (G) into two cliques. Thus, N [v1 ], . . . , N [vn ] is a strictly connected path passing through all hyperedges of N [G]. Now we are ready to prove the rigidity result for the closed neighborhood hypergraphs of PCA graphs. As discussed in Section 1, the present proof of Part 1 of the following theorem is alternative to the theory of local tournament orientations [12]. Theorem 3.3. Let G be a twin-free, connected PCA graph with more than one vertex. 1. If G is non-bipartite, then N [G] has a unique arc ordering (up to reversal). 2. If G is bipartite and connected, then N [G] has exactly two tight arc orderings (up to reversal). Proof. 1. The existence of an arc ordering of N [G] is a simple fact stated in Part 1 of Lemma 3.1. By Part 2 of Lemma 3.1, any arc ordering of N [G] is tight. The uniqueness follows from Theorem 1.3.2 by Theorem 3.2. 2. Recall that the open neighborhood hypergraph of a graph G is defined by N (G) = {N (v)}v∈V (G) . In place of N [G], it is now practical to consider the complement hypergraph N [G] = V (G) \ N [v] v∈V (G) . Note that N [G] = N (G). This hypergraph is disconnected, and any tight arc ordering of N [G] induces a tight interval ordering of each connected component of N (G). By Lemma 3.1.1, this applies to any geometric ordering of N [G]. Conversely, arbitrary tight interval orderings of the components of N (G) can be merged into a tight arc ordering of N [G]. Since G is connected, N (G) has exactly two components. Applying Theorem 1.3.1 to each of them, we conclude that each of the two components has a single tight interval ordering (up to reversal). Since there are exactly two essentially different ways to merge them, we see that N [G] has exactly two tight arc orderings (up to reversal).
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Fig. 1. An example: (a) A proper arc system; (b) The corresponding intersection graph G. Its complement G is non-bipartite; (c) The closed neighborhood hypergraph N [G] is not strictly overlap-connected: the hyperedge N [a] forms a single strictly overlap-connected component.
Fig. 2. A non-co-bipartite, twin-free, connected PCA graph on n vertices can have more than n/3 vertices of degree n − 2 (a vertex v has degree n − 2 iff |N [v]| = n − 1). For each k ≥ 2, define a graph Gk on 3k − 1 vertices by its arc model Ak such that k of the vertices will have degree n − 2. On the circle C = {u1 , u2 , . . . , uk , v1 , w1 , v2 , w2 , . . . , vk−1 , wk−1 , vk }, whose points go in this circular order, consider arcs U1 = [u1 , uk−1 ], Ui = [ui , vi−1 ] for 2 ≤ i ≤ k, V1 = [v1 , vk ], Vi = [vi , ui−1 ] = C \ Ui for 2 ≤ i ≤ k, and Wi = Vi \ {vi } for i ≤ k − 1. As Ak is tight, it can easily be made proper and, hence, Gk is PCA. Since the arcs have pairwise different left endpoints, we can identify V (Gk ) = C. The graph can be described by listing the pairs of non-adjacent vertices, namely vi ui , wi ui , wi ui+1 , and u1 uk . There are no twins, and Gk is not co-bipartite because its complement contains an odd cycle, namely u1 w1 u2 w2 . . . uk . In accordance with Theorem 3.4, when we remove from N [Gk ] the hyperedges N [vi ] = C \ {ui }, the hypergraph stays twinfree and becomes strictly overlap-connected. This can be seen by looking at the following path in the complemented hypergraph: N [u1 ] = {uk , v1 , w1 }, N [u2 ] = {w1 , v2 , w2 }, . . . , N [uk−1 ] = {wk−2 , vk−1 , wk−1 }, N [uk ] = {wk−1 , vk , u1 }, N [w1 ] = {u1 , u2 }, . . . , N [wk−1 ] = {uk−1 , uk }. The figure shows A3 and an arc model for N [G3 ] which has the arcs of size n − 1 grayed out.
Let us stress that Part 2 of Theorem 3.3 concerns only tight orderings. To show that it cannot be strengthened to the class of all arc orderings, consider the half-graph Hm that is the bipartite graph with vertex classes {u1 , . . . , um } and {v1 , . . . , vm } where ui is adjacent to vj if i ≤ j. The complement Gm = Hm is a twin-free, connected PCA graph. Note that, besides two pairs of mutually reversed tight interval orderings, the hypergraph N (H3 ) has another interval ordering and, hence, N [G3 ] has a non-tight arc ordering. If we increase the parameter m, the number of non-tight arc orderings of N [Gm ] grows exponentially. Let G be a twin-free, connected PCA graph with non-bipartite complement. In Theorem 3.2 we established that N [G] is strictly connected. We now take a closer look at the connectedness properties of this hypergraph. Note that N [G] is not always strictly overlap-connected: It can have hyperedges of size n − 1, and each such hyperedge forms a separate strictly overlap-connected component; see an example in Fig. 1. Nevertheless, if we remove the (n − 1)-element hyperedge from N [G] in this example, the remaining hypergraph becomes strictly overlap-connected and stays twin-free. It turns out that this is a general phenomenon. Theorem 3.4. If G is a twin-free, non-co-bipartite, connected PCA graph on n vertices, then the hypergraph obtained from N [G] by removing all hyperedges of size n − 1 is twin-free and strictly overlap-connected. Proof. Assume that a graph G satisfies all the assumptions. Note that then G has n ≥ 4 vertices. Denote H = N [G]. Since G is connected, H has no hyperedge of size 1. It also has no hyperedge of size n because G is non-co-bipartite and, hence, has no universal vertex. Remove all hyperedges of size n − 1 and denote the result by H ′ . By Theorem 3.3.1, H has a unique arc ordering (up to reversal). Therefore, H ′ is twin-free and strictly overlap-connected by Theorem 1.2. Note that Theorem 3.4 gives us a complete description of the decomposition of N [G] into strictly overlap-connected components because each hyperedge of size n − 1 forms such a component alone. Fig. 2 shows that the number of such single-hyperedge components can be linear.
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Fig. 3. Proof of Theorem 3.5.
We conclude with pointing out yet another application of the results of Section 2. Theorem 3.5 is a recasting of the well-known consequence of [12, Theorem 4.5] that any twin-free, connected PCA graph G with non-bipartite or connected complement has a unique proper arc representation. We show that Theorem 1.3 provides an alternative, rather short way of proving this important fact. Theorem 3.5. Let G be a twin-free, connected PCA graph. If its complement G is non-bipartite or connected, then all geometric orders of N [G] associated with proper arc representations of G are equal up to reversing. Proof. If G is non-bipartite, the result follows from Part 1 of Theorem 3.3. If G is bipartite and connected, Part 2 of Theorem 3.3 leaves two different possibilities. It turns out that, nevertheless, a geometric order is unique also in this case. Let V (G) = U ∪ W be the bipartition of G into two independent sets. Note that U and W span two connected components of the hypergraph N (G). Consider an arbitrary proper arc representation α of G and the corresponding geometric order ≺α on V (G). Like in the proof of Theorem 3.3.2, notice that ≺α is a tight interval order for each connected component of N (G) and, therefore, the restrictions of ≺α to U and W , each considered up to reversing, do not depend on α by Theorem 1.3.1. This still leaves two distinct possibilities of merging them into ≺α . We now show that one of them is actually ruled out, and this will prove that ≺α does not depend on α , if this order is considered up to reversing. Note that, since G is connected, G cannot have universal vertices. In this case it is well known and easy to see [13, Section 4.2] that any proper arc representation α of G has the following property: If v and v ′ are adjacent vertices of G, then the arcs α(v) and α(v ′ ) strictly overlap, that is, contain exactly one endpoint of each other. The assumptions of the theorem imply that both U and W contain at least two vertices. Let w and w ′ be two vertices in W . Since w and w ′ are not twins, they are distinguished by adjacency to some vertex u ∈ U. W.l.o.g. suppose that w and u are adjacent, while w ′ and u are not. Since w is not universal in G, there is a vertex u′ in U non-adjacent to w . Note that α(w ′ ) and α(u) contain different endpoints of α(w), and α(u′ ) and α(w) contain different endpoints of α(u); see Fig. 3. It follows that the arcs α(w ′ ), α(w), α(u), and α(u′ ) occur in the arc model exactly in this circular order, irrespective of whether α(w ′ ) and α(u′ ) intersect or not. Since the quadruple w ′ , w, u, u′ was chosen in terms of the graph G alone and α was supposed to be arbitrary, this conclusion holds true for any proper arc representation of G. Therefore, there is a unique way of merging the restrictions of ≺α to U and W so as to obtain an ordering of V (G) consistent with some geometric order. This proves the desired uniqueness result. Acknowledgments The first two authors are supported by DFG grant KO 1053/7–2. The third author is supported by DFG grant VE 652/1–1. We are grateful to the anonymous referee who provided us with numerous detailed comments and useful references to the earlier work on PCA graphs. We also thank another anonymous referee whose suggestion to extend the approach of Chen and Yesha [4, Theorem 2] to CA hypergraphs led to simplifying our original proof of Lemma 2.1. References [1] J. Bang-Jensen, J. Huang, A. Yeo, Convex-round and concave-round graphs, SIAM J. Discrete Math. 13 (2) (2000) 179–193. [2] K. Booth, G. Lueker, Testing for the consecutive ones property, interval graphs, and graph planarity using PQ -tree algorithms, J. Comput. System Sci. 13 (3) (1976) 335–379. [3] E. Boros, V. Gurvich, I.E. Zverovich, Neighborhood hypergraphs of bipartite graphs, J. Graph Theory 58 (1) (2008) 69–95. [4] L. Chen, Y. Yesha, Parallel recognition of the consecutive ones property with applications, J. Algorithms 12 (3) (1991) 375–392. [5] A.R. Curtis, M.C. Lin, R.M. McConnell, Y. Nussbaum, F.J. Soulignac, J.P. Spinrad, J.L. Szwarcfiter, Isomorphism of graph classes related to the circular-ones property, Discrete Math. Theor. Comput. Sci. 15 (1) (2013) 157–182. [6] X. Deng, P. Hell, J. Huang, Linear-time representation algorithms for proper circular-arc graphs and proper interval graphs, SIAM J. Comput. 25 (2) (1996) 390–403. [7] M. Dom, Algorithmic aspects of the consecutive-ones property, Bull. EATCS 98 (2009) 27–59. [8] F. Gavril, R.Y. Pinter, S. Zaks, Intersection representations of matrices by subtrees and unicycles on graphs, J. Discrete Algorithms 6 (2) (2008) 216–228. [9] P. Hell, J. Huang, Lexicographic orientation and representation algorithms for comparability graphs, proper circular arc graphs, and proper interval graphs, J. Graph Theory 20 (3) (1995) 361–374.
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