Boxicity and Poset Dimension
aa r X i v : . [ m a t h . C O ] M a r Boxicity and Poset Dimension
Abhijin Adiga , Diptendu Bhowmick , L. Sunil Chandran Department of Computer Science and Automation, Indian Institute of Science, Bangalore–560012, India.emails: [email protected], [email protected], [email protected]
Abstract.
Let G be a simple, undirected, finite graph with vertex set V ( G ) and edge set E ( G ). A k -dimensional box is a Cartesian product of closed intervals [ a , b ] × [ a , b ] × · · · × [ a k , b k ]. The boxicity of G ,box( G ) is the minimum integer k such that G can be represented as the intersection graph of k -dimensionalboxes, i.e. each vertex is mapped to a k -dimensional box and two vertices are adjacent in G if and only iftheir corresponding boxes intersect. Let P = ( S, P ) be a poset where S is the ground set and P is a reflexive,anti-symmetric and transitive binary relation on S . The dimension of P , dim( P ) is the minimum integer t such that P can be expressed as the intersection of t total orders.Let G P be the underlying comparability graph of P , i.e. S is the vertex set and two vertices are adjacent if andonly if they are comparable in P . It is a well-known fact that posets with the same underlying comparabilitygraph have the same dimension. The first result of this paper links the dimension of a poset to the boxicityof its underlying comparability graph. In particular, we show that for any poset P , box( G P ) / ( χ ( G P ) − ≤ dim( P ) ≤ G P ), where χ ( G P ) is the chromatic number of G P and χ ( G P ) = 1. It immediately followsthat if P is a height-2 poset, then box( G P ) ≤ dim( P ) ≤ G P ) since the underlying comparability graphof a height-2 poset is a bipartite graph.The second result of the paper relates the boxicity of a graph G with a natural partial order associated withthe extended double cover of G , denoted as G c : Note that G c is a bipartite graph with partite sets A and B which are copies of V ( G ) such that corresponding to every u ∈ V ( G ), there are two vertices u A ∈ A and u B ∈ B and { u A , v B } is an edge in G c if and only if either u = v or u is adjacent to v in G . Let P c bethe natural height-2 poset associated with G c by making A the set of minimal elements and B the set ofmaximal elements. We show that box ( G )2 ≤ dim( P c ) ≤ G ) + 4.These results have some immediate and significant consequences. The upper bound dim( P ) ≤ G P )allows us to derive hitherto unknown upper bounds for poset dimension such as dim( P ) ≤ G P )+4, since boxicity of any graph is known to be at most its tree-width + 2. In the other direction, using thealready known bounds for partial order dimension we get the following: (1) The boxicity of any graph withmaximum degree ∆ is O ( ∆ log ∆ ) which is an improvement over the best known upper bound of ∆ + 2.(2) There exist graphs with boxicity Ω ( ∆ log ∆ ). This disproves a conjecture that the boxicity of a graph is O ( ∆ ). (3) There exists no polynomial-time algorithm to approximate the boxicity of a bipartite graph on n vertices with a factor of O ( n . − ǫ ) for any ǫ >
0, unless NP = ZP P . Keywords:
Boxicity, partial order, poset dimension, comparability graph, extended double cover. A k -box is a Cartesian product of closed intervals [ a , b ] × [ a , b ] × · · · × [ a k , b k ]. A k -box representation of agraph G is a mapping of the vertices of G to k -boxes in the k -dimensional Euclidean space such that two vertices This work was supported by DST grant SR/S3/EECE/62/2006 and Infosys Fellowship. n G are adjacent if and only if their corresponding k -boxes have a non-empty intersection. The boxicity of a graphdenoted box( G ), is the minimum integer k such that G has a k -box representation. Boxicity was introduced byRoberts [24]. Cozzens [9] showed that computing the boxicity of a graph is NP-hard. This was later strengthenedby Yannakakis [31] and finally by Kratochv`ıl [22] who showed that determining whether boxicity of a graph is atmost two itself is NP-complete.It is easy to see that a graph has boxicity at most 1 if and only if it is an interval graph , i.e. each vertex of thegraph can be associated with a closed interval on the real line such that two intervals intersect if and only if thecorresponding vertices are adjacent. By definition, boxicity of a complete graph is 0. Let G be any graph and G i ,1 ≤ i ≤ k be graphs on the same vertex set as G such that E ( G ) = E ( G ) ∩ E ( G ) ∩ · · · ∩ E ( G k ). Then we saythat G is the intersection graph of G i s for 1 ≤ i ≤ k and denote it as G = T ki =1 G i . Boxicity can be stated interms of intersection of interval graphs as follows: Lemma 1.
Roberts [24]:
The boxicity of a non-complete graph G is the minimum positive integer b such that G can be represented as the intersection of b interval graphs. Moreover, if G = T mi =1 G i for some graphs G i then box( G ) ≤ P mi =1 box( G i ) . Roberts, in his seminal work [24] proved that the boxicity of a complete k -partite graph is k . Chandranand Sivadasan [6] showed that box( G ) ≤ tree-width ( G ) + 2. Chandran, Francis and Sivadasan [5] proved thatbox( G ) ≤ χ ( G ) where, χ ( G ) is the chromatic number of G . In [14] Esperet proved that box( G ) ≤ ∆ ( G ) + 2,where ∆ ( G ) is the maximum degree of G . Scheinerman [25] showed that the boxicity of outer planar graphs isat most 2. Thomassen [26] proved that the boxicity of planar graphs is at most 3. In [11], Cozzens and Robertsstudied the boxicity of split graphs. A partially ordered set or poset P = ( S, P ) consists of a non empty set S , called the ground set and a reflexive,antisymmetric and transitive binary relation P on S . A total order is a partial order in which every two elementsare comparable. It essentially corresponds to a permutation of elements of S . A height- poset is one in whichevery element is either a minimal element or a maximal element. A linear extension L of a partial order P is atotal order which satisfies ( x ≤ y in P ⇒ x ≤ y in L ). A realizer of a poset P = ( S, P ) is a set of linear extensionsof P , say R which satisfy the following condition: for any two distinct elements x and y , x < y in P if and only if x < y in L , ∀ L ∈ R . The poset dimension of P (sometimes abbreviated as dimension of P ) denoted by dim( P ) isthe minimum integer k such that there exists a realizer of P of cardinality k . Poset dimension was introduced byDushnik and Miller [12]. Clearly, a poset is one-dimensional if and only if it is a total order. Pnueli et al. [23] gavea polynomial time algorithm to recognize dimension 2 posets. In [31] Yannakakis showed that it is NP-completeto decide whether the dimension of a poset is at most 3. For more references and survey on dimension theoryof posets see Trotter [28,29]. Recently, Hegde and Jain [21] showed that it is hard to design an approximationalgorithm for computing the dimension of a poset.A simple undirected graph G is a comparability graph if and only if there exists some poset P = ( S, P ), suchthat S is the vertex set of G and two vertices are adjacent in G if and only if they are comparable in P . We willcall such a poset an associated poset of G . Likewise, we refer to G as the underlying comparability graph of P .Note that for a height-2 poset, the underlying comparability graph is a bipartite graph with partite sets A and B , with say A corresponding to minimal elements and B to maximal elements. For more on comparability graphssee [19]. It is easy to see that there is a unique comparability graph associated with a poset, whereas, there can beseveral posets with the same underlying comparability graph. However, Trotter, Moore and Sumner [30] provedthat posets with the same underlying comparability graph have the same dimension.2 Our Main Results
The results of this paper are the consequence of our attempts to bring out some connections between boxicityand poset dimension. As early as 1982, Yannakakis had some intuition regarding a possible connection betweenthese problems when he established the NP-completeness of both poset dimension and boxicity in [31]. Butinterestingly, no results were discovered in the last 25 years which establish links between these two notions.Perhaps the researchers were misled by some deceptive examples such as the following one: Consider a completegraph K n where n is even and remove a perfect matching from it. The resulting graph is a comparability graphand the dimension of any of its associated posets is 2, while its boxicity is n/
2. In this context it may be worthrecalling a result from [16] which relates the poset dimension to another parameter namely the dimension of boxorders. A poset P = ( S, P ) is said to be a box order in m dimensions if there exists a mapping of its elementsto m -dimensional axis-parallel boxes such that x < y in P if and only if the box of y strictly contains the boxof x . Box order is a particular type of geometrical containment order (See [16,28]). The result is as follows: thedimension of P is at most 2 m if and only if it is a box order in m dimensions [18,20]. But note that boxicity isfundamentally different from box orders. As in the case of the above example, we can demonstrate families ofposets of constant dimension whose underlying comparability graphs have arbitrarily high boxicity, which is incontrast with the above result on box orders.First we state an upper bound and a lower bound for the dimension of a poset in terms of the boxicity of itsunderlying comparability graph. Theorem 1.
Let P = ( V, P ) be a poset such that dim( P ) > and G P its underlying comparability graph. Then, dim( P ) ≤ G P ) . Theorem 2.
Let P = ( V, P ) be a poset and let χ ( G P ) be the chromatic number of its underlying comparabilitygraph G P such that χ ( G P ) > . Then, dim( P ) ≥ box ( G P ) χ ( G P ) − . Note that if P is a height-2 poset, then G P is a bipartite graph and therefore χ ( G P ) = 2. Thus, from the aboveresults we have the following: Corollary 1.
Let P = ( V, P ) be a height- poset and G P its underlying comparability graph. Then, box( G P ) ≤ dim( P ) ≤ G P ) . The double cover and extended double cover of a graph are popular notions in graph theory. They provide a naturalway to associate a bipartite graph to the given graph. In this paper we make use of the latter construction.
Definition 1.
The extended double cover of G , denoted as G c is a bipartite graph with partite sets A and B which are copies of V ( G ) such that corresponding to every u ∈ V ( G ) , there are two vertices u A ∈ A and u B ∈ B and { u A , v B } is an edge in G c if and only if either u = v or u is adjacent to v in G . We prove the following lemma relating the boxicity of G and G c . Lemma 2.
Let G be any graph and G c its extended double cover. Then, box( G )2 ≤ box( G c ) ≤ box( G ) + 2 . Let P c be the natural height-2 poset associated with G c , i.e. the elements in A are the minimal elements and theelements in B are the maximal elements. Combining Corollary 1 and Lemma 2 we have the following theorem: Theorem 3.
Let G be a graph and P c be the natural height- poset associated with its extended double cover.Then, dim( P c )2 − ≤ box( G ) ≤ P c ) and therefore box( G ) = Θ (dim( P c )) . .1 ConsequencesNew upper bounds for poset dimension: Our results lead to some hitherto unknown bounds for posetdimension. Some general bounds obtained in this manner are listed below:1. It is proved in [6] that for any graph G , boxicity of G is at most tree-width ( G ) + 2. For more information on tree-width see [2]. Applying this bound in Theorem 1 it immediately follows that, for a poset P , dim( P ) ≤ G P ) + 4.2. The threshold dimension of a graph G is the minimum number of threshold graphs such that G is the edgeunion of these graphs. For more on threshold graphs and threshold dimension see [19]. Cozzens and Halsey[10] proved that box( G ) ≤ threshold-dimension( G ), where G is the complement of G . From this it followsthat dim( P ) ≤ G P ).3. In [3] it is proved that box( G ) ≤ (cid:22) MVC( G ) (cid:23) + 1, where MVC ( G ) is the cardinality of the minimum vertexcover of G . Therefore, we have dim( P ) ≤ MVC ( G P ) + 2.Some more interesting results can be obtained if we restrict G P to belong to certain subclasses of graphs. Notethat there are several research papers in the partial order literature which study the dimension of posets whoseunderlying comparability graph has some special structure – interval order, semi order and crown posets are someexamples.4. Scheinerman [25] proved that the boxicity of outer planar graphs is at most 2 and later Thomassen [26] provedthat the boxicity of planar graphs is at most 3. Therefore, it follows that dim( P ) ≤ G P is outer planarand dim( P ) ≤ G P is planar.5. Bhowmick and Chandran [1] proved that boxicity of AT-free graphs is at most χ ( G P ). Hence, dim( P ) ≤ χ ( G P ), if G P is AT-free.6. If G P is an interval graph, then, we get from Theorem 1, dim( P ) ≤
2, since box( G P ) = 1. However, observingthat interval graphs are co-comparability graphs this result would follow also as a consequence of a result byDushnik and Miller [12]: dim( P ) ≤ G P is a co-comparability graph.7. The boxicity of a d - dimensional hypercube is O ( d/ log( d )) [7]. Therefore, if G P is a height-2 poset whichcorresponds to a d - dimensional hypercube , then from Corollary 1 we have dim( P ) = O ( d/ log( d )).8. Chandran et al. [4] recently proved that chordal bipartite graphs have arbitrarily high boxicity. From Corollary1 it follows that height-2 posets whose underlying comparability graph are chordal bipartite graphs can havearbitrarily high dimension. Improved upper bound for boxicity based on maximum degree:
F¨uredi and Kahn [17] proved thefollowing
Lemma 3.
Let P be a poset and ∆ be the maximum degree of G P . Then, there exists a constant c such that dim( P ) < c∆ (log ∆ ) . From Lemma 2 and Corollary 1 we have box( G ) ≤ G c ) ≤ P c ), where G c is the extended double coverof G . Note that by construction ∆ ( G c ) = ∆ ( G ) + 1. On applying the above lemma, we have Theorem 4.
For any graph G having maximum degree ∆ there exists a constant c ′ such that box( G ) < c ′ ∆ (log ∆ ) . This result is an improvement over the previous upper bound of ∆ + 2 by Esperet [14].4 ounter examples to the conjecture of [5]: Chandran et al. [5] conjectured that boxicity of a graph is O ( ∆ ).We use a result by Erd˝os, Kierstead and Trotter [13] to show that there exist graphs with boxicity Ω ( ∆ log ∆ ),hence disproving the conjecture. Let P ( n, p ) be the probability space of height-2 posets with n minimal elementsforming set A and n maximal elements forming set B , where for any a ∈ A and b ∈ B , Prob( a < b ) = p ( n ) = p .They proved the following: Theorem 5. [13]
For every ǫ > , there exists δ > so that if (log ǫ n ) /n < p < − n − ǫ , then, dim( P ) > ( δpn log( pn )) / (1 + δp log( pn )) for almost all P ∈ P ( n, p ) . When p = 1 / log n , for almost all posets P ∈ P ( n, / log n ), ∆ ( G P ) < δ n/ log n and by the above theoremdim( P ) > δ n , where δ and δ are some positive constants (see [29] for a discussion on the above theorem). FromTheorem 1, it immediately implies that for almost all P ∈ P ( n, / log n ), box( G P ) ≥ dim( P )2 > δ ′ ∆ ( G P ) log ∆ ( G P )for some positive constant δ ′ , hence proving the existence of graphs with boxicity Ω ( ∆ log ∆ ). Approximation hardness for the boxicity of bipartite graphs:
Hegde and Jain [21] proved the following
Theorem 6.
There exists no polynomial-time algorithm to approximate the dimension of an n -element posetwithin a factor of O ( n . − ǫ ) for any ǫ > , unless N P = ZP P . This is achieved by reducing the fractional chromatic number problem on graphs to the poset dimension problem.In addition they observed that a slight modification of their reduction will imply the same result for even height-2posets. From Corollary 1, it is clear that for any height-2 poset P , dim( P ) = Θ (box( G P )). Suppose there exists analgorithm to compute the boxicity of bipartite graphs with approximation factor O ( n . − ǫ ), for some ǫ >
0, then,it is clear that the same algorithm can be used to compute the dimension of height-2 posets with approximationfactor O ( n . − ǫ ), a contradiction. Hence, Theorem 7.
There exists no polynomial-time algorithm to approximate the boxicity of a bipartite graph on n -vertices with a factor of O ( n . − ǫ ) for any ǫ > , unless N P = ZP P . Let [ n ] denote { , , . . . , n } where n is a positive integer. For any graph G , let V ( G ) and E ( G ) denote its vertexset and edge set respectively. If G is undirected, for any u, v ∈ V ( G ), { u, v } ∈ E ( G ) means u is adjacent to v andif G is directed, ( u, v ) ∈ E ( G ) means there is a directed edge from u to v . Whenever we refer to an AB bipartite(or co-bipartite) graph, we imply that its vertex set is partitioned into non-empty sets A and B where both thesesets induce independent sets (cliques respectively).In a poset P = ( S, P ), the notations aP b , a ≤ b in P and ( a, b ) ∈ P are equivalent and are used interchangeably. G P denotes the underlying comparability graph of P . A subset of P is called a chain if each pair of distinctelements is comparable. If each pair of distinct elements is incomparable, then it is called an antichain . Given an AB bipartite graph G , the natural poset associated with G with respect to the bipartition is the poset obtainedby taking A to be the set of minimal elements and B to be the set of maximal elements. In particular, if G c isthe extended double cover of G , we denote by P c the natural associated poset of G c .Suppose I is an interval graph. Let f I be an interval representation for I , i.e. it is a mapping from the vertexset to closed intervals on the real line such that for any two vertices u and v , { u, v } ∈ E ( I ) if and only if f I ( u ) ∩ f I ( v ) = ∅ . Let l ( u, f I ) and r ( u, f I ) denote the left and right end points of the interval corresponding tothe vertex u respectively. In this paper, we will never consider more than one interval representation for an interval5raph. Therefore, we will simplify the notations to l ( u, I ) and r ( u, I ). Further, when there is no ambiguity aboutthe graph under consideration and its interval representation, we simply denote the left and right end points as l ( u ) and r ( u ) respectively. Note that for any interval graph there exists an interval representation with all endpoints distinct. Such a representation is called a distinguishing interval representation. It is an easy exercise toderive such a distinguishing interval representation starting from an arbitrary interval representation of the graph. Let box( G P ) = k . Note that since dim( P ) > G P cannot be a complete graph and therefore k ≥
1. Let I = { I , I , . . . , I k } be a set of interval graphs such that G P = T ki =1 I i . Now, corresponding to each I i we willconstruct two total orders L i and L i such that R = { L ji | i ∈ [ k ] and j ∈ [2] } is a realizer of P .Let I ∈ I and f I be an interval representation of I . We will define two partial orders P I and P I as follows: ∀ a ∈ V , ( a, a ) belongs to P I and P I and for every non-adjacent pair of vertices a, b ∈ V with respect to I ,( a, b ) ∈ P I ( b, a ) ∈ P I ) if and only if r ( a, f I ) < l ( b, f I ) . Partial orders constructed in the above manner from a collection of closed intervals are called interval orders (See[29] for more details). It is easy to see that I (the complement of I ) is the underlying comparability graph of both P I and P I .Let G and G be two directed graphs with vertex set V and edge set E ( G ) = ( P ∪ P I ) \ { ( a, a ) | a ∈ V } and E ( G ) = ( P ∪ P I ) \ { ( a, a ) | a ∈ V } respectively. Note that from the definition it is obvious that there are nodirected loops in G and G . Lemma 4. G and G are acyclic directed graphs.Proof. We will prove the lemma for G – a similar proof holds for G . First of all, since G P is not a completegraph P I = ∅ . Suppose P I is a total order, i.e. if P is an antichain, then it is clear that E ( G ) = P I and therefore G is acyclic. Henceforth, we will assume that P I is not a total order.Suppose G is not acyclic. Let C = { ( a , a ) , ( a , a ) , . . . , ( a t − , a t − ) , ( a t − , a ) } be a shortest directed cyclein G .First we will show that t > t is the length of C ). If t = 2, then there should be a, b ∈ V such that( a, b ) , ( b, a ) ∈ E ( G ). Since P is a partial order, ( a, b ) and ( b, a ) cannot be simultaneously present in P . The sameholds for P I . Thus, without loss of generality we can assume that ( a, b ) ∈ P and ( b, a ) ∈ P I . But if ( a, b ) ∈ P ,then, a and b are adjacent in G P and thus adjacent in I . Then clearly the intervals of a and b intersect andtherefore ( b, a ) / ∈ P I , a contradiction.Now, we claim that two consecutive edges in C cannot belong to P (or P I ). Suppose there do exist such edges,say ( a i , a i +1 ) and ( a i +1 , a i +2 ) which belong to P (or P I ) (note that the addition is modulo t ). Since P (or P I ) isa partial order, it implies that ( a i , a i +2 ) ∈ P (or P I ) and as a result we have a directed cycle of length t −
1, acontradiction to the assumption that C is a shortest directed cycle. Therefore, the edges of C alternate between P and P I . It also follows that t ≥ a , a ) , ( a , a ) ∈ P I . We claim that { ( a , a ) , ( a , a ) } is aninduced poset of P I . First of all a and a are not comparable in P I as they are comparable in P . If either { a , a } or { a , a } are comparable, then we can demonstrate a shorter directed cycle in G , a contradiction.Finally we consider the pair { a , a } . If t = 4, then they are not comparable as they are comparable in P whileif t = 4 and if they are comparable, then, it would again imply a shorter directed cycle, a contradiction. Hence,6 ( a , a ) , ( a , a ) } is an induced subposet. In the literature such a poset is denoted as + where + refers to disjoint sum and is a two-element total order. Fishburn [15] has proved that a poset is an interval order if andonly if it does not contain a + . This implies that P I is not an interval order, a contradiction.We have therefore proved that there cannot be any directed cycles in G . In a similar way we can show that G is an acyclic directed graph. ⊓⊔ Since G and G are acyclic, we can construct total orders, say L and L using topological sort on G and G such that P ∪ P I ⊆ L and P ∪ P I ⊆ L (For more details on topological sort, see [8] for example).For each I i , we create linear extensions L i and L i as described above. We claim that R = { L ji | i ∈ [ k ] , j ∈ [2] } is a realizer of P . For each L ji , it is clear from construction that P ⊂ L ji . If a and b are not comparable in P , then { a, b } / ∈ E ( G P ), and therefore there exists some interval graph I q ∈ I such that { a, b } / ∈ E ( I q ). Assuming thatthe interval for a occurs before the interval for b in the interval representation of I q , it follows by constructionthat ( a, b ) ∈ P Iq and ( b, a ) ∈ P Iq and therefore ( a, b ) ∈ L q and ( b, a ) ∈ L q . Hence proved. Consider the crown poset S n : a height-2 poset with n minimal elements a , a , . . . , a n and n maximal elements b , b , . . . , b n and a i < b j , for j = i +1 , i +2 , . . . , i −
1, where the addition is modulo n . Its underlying comparabilitygraph is the bipartite graph obtained by removing a perfect matching from the complete bipartite graph K n,n .The dimension of this poset is n (see [27,29]) while the boxicity of the graph is (cid:6) n (cid:7) [3]. We will prove that box( G P ) ≤ ( χ ( G P ) −
1) dim( P ). Let ( χ ( G P ) −
1) = p , dim( P ) = k and R = { L , . . . , L k } a realizer of P . Now we color the vertices of G P as follows: For a vertex v , if γ is the length of a longest chainin P such that v is its maximum element, then we assign color γ to it. This is clearly a proper coloring schemesince if two vertices x and y are assigned the same color, say γ and x < y , then it implies that the length of alongest chain in which y occurs as the maximum element is at least γ + 1, a contradiction. Also, this is a minimumcoloring because the maximum number that gets assigned to any vertex equals the length of a longest chain in P , which corresponds to the clique number of G P .Now we construct pk interval graphs I = { I ij | i ∈ [ p ] , j ∈ [ k ] } and show that G P is an intersection graph ofthese interval graphs. Let Π j be the permutation induced by the total order L j on [ n ], i.e. xL j y if and only if Π − j ( x ) < Π − j ( y ). The following construction applies to all graphs in I except I pk . Let I ij ∈ I \ { I pk } . Weassign the point interval (cid:2) Π − j ( v ) , Π − j ( v ) (cid:3) for all vertices v colored i . For all vertices v colored i ′ < i , we assign (cid:2) Π − j ( v ) , n + 1 (cid:3) and for those colored i ′ > i , we assign (cid:2) , Π − j ( v ) (cid:3) . The interval assignment for the last intervalgraph I pk is as follows: for all vertices v colored p + 1 = χ ( G P ) we assign the point interval (cid:2) Π − k ( v ) , Π − k ( v ) (cid:3) and for the rest of the vertices we assign the interval (cid:2) Π − k ( v ) , n + 1 (cid:3) . Next, we will show that G P = T I ∈I I . Claim 1. If u and v are adjacent in G P , then they are adjacent in all I ∈ I . Proof.
Let u be colored i and v be colored i ′ . It is clear that i = i ′ and without loss of generality we will assumethat i < i ′ . By the way we have colored, it implies that u < v in P and therefore Π − j ( u ) < Π − j ( v ), ∀ j ∈ [ k ]. Let I hj , h ∈ [ p ] and j ∈ [ k ] be the interval graph under consideration. There are 5 possible cases which we considerone by one: 7 ase 1: ( h < i, i ′ ) By construction in I hj , u and v are assigned intervals (cid:2) , Π − j ( u ) (cid:3) and (cid:2) , Π − j ( v ) (cid:3) respectivelyand therefore u and v are adjacent in I hj , ∀ j ∈ [ k ]. Case 2: ( i, i ′ < h ) u and v are assigned intervals (cid:2) Π − j ( u ) , n + 1 (cid:3) and (cid:2) Π − j ( v ) , n + 1 (cid:3) respectively and thereforeare adjacent in I hj , ∀ j ∈ [ k ]. Case 3: ( i < h < i ′ ) u is assigned interval (cid:2) Π − j ( u ) , n + 1 (cid:3) and v is assigned interval (cid:2) , Π − j ( v ) (cid:3) . Since 0 <Π − j ( u ) < Π − j ( v ) < n + 1, it follows that u is adjacent to v in I hj , ∀ j ∈ [ k ]. Case 4: ( h = i ) If h = p and j = k , then i ′ = p + 1 and therefore u is assigned (cid:2) Π − k ( u ) , n + 1 (cid:3) and v is assigned (cid:2) Π − k ( v ) , Π − k ( v ) (cid:3) . If not, then u is assigned the point interval (cid:2) Π − j ( u ) , Π − j ( u ) (cid:3) and v is assigned (cid:2) , Π − j ( v ) (cid:3) .In either case, since Π − j ( u ) < Π − j ( v ), the two vertices are adjacent. Case 5: ( h = i ′ ) Since h ≤ p = χ ( G P ) −
1, it implies that i, i ′ ≤ p . Therefore, if h = p and j = k , then u and v are assigned (cid:2) Π − j ( u ) , n + 1 (cid:3) and (cid:2) Π − j ( v ) , n + 1 (cid:3) respectively. If not, then v is assigned the point interval (cid:2) Π − j ( v ) , Π − j ( v ) (cid:3) and u is assigned (cid:2) Π − j ( u ) , n + 1 (cid:3) . Again, since Π − j ( u ) < Π − j ( v ), in either case the twovertices are adjacent. Hence proved. Claim 2. If u and v are not adjacent in G P , then there exists some I ∈ I such that { u, v } / ∈ E ( I ). Proof.
Again let u be colored i and v be colored i ′ . Recall that k ≥
2. If i = i ′ , then by construction it is clearthat u and v are not adjacent in I i if i = p + 1 and when i = p + 1, then they are not adjacent in I pk . Therefore,without loss of generality we will assume that i < i ′ . Since u and v are not adjacent in G P , they are incomparablein P and therefore, there exists some l ∈ [ k ] such that u > v in L l which in turn implies that Π − l ( u ) > Π − l ( v ).There are 2 possible cases: Case 1: ( i < p ) Since i < i ′ , in I il , u and v are assigned intervals (cid:2) Π − l ( u ) , Π − l ( u ) (cid:3) and (cid:2) , Π − l ( v ) (cid:3) respectivelyand therefore, since Π − l ( u ) > Π − l ( v ) u and v are not adjacent in I il . Case 2: ( i = p ) Clearly i ′ = p + 1. If l < k , then it is similar to the previous case. If l = k , then, in I pk , u and v areassigned (cid:2) Π − k ( u ) , n + 1 (cid:3) and (cid:2) Π − k ( v ) , Π − k ( v ) (cid:3) respectively. Since Π − l ( u ) > Π − l ( v ), u and v are not adjacentin I pk .Hence we have proved Theorem 2.Consider a complete k -partite graph G on n = qk vertices where q, k >
1, i.e. V ( G ) = V ⊎ V ⊎ · · · ⊎ V k is apartition of V ( G ) where | V i | = q . For any two vertices x ∈ V i and y ∈ V j , { x, y } ∈ E ( G ) if and only if i = j . G isa comparability graph and here is one transitive orientation of G : for every pair of adjacent vertices u ∈ V i and v ∈ V j , where u, v ∈ [ k ] and i = j , make u < v if and only if i < j . Let P be the resulting poset. It is an easyexercise to show that dim( P ) = 2. The chromatic number of G is k and Roberts [24] showed that its boxicity is k .From Theorem 2 it follows that dim( P ) ≥ kk − . Therefore, the complete k -partite graph serves as a tight examplefor Theorem 2.However, it would be interesting to see if there are posets of higher dimension for which Theorem 2 is tight. In this section, we will prove Lemma 2. But first, we will need some definitions and lemmas.8 efinition 2.
Let H be an AB bipartite graph. The associated co-bipartite graph of H , denoted by H ∗ is thegraph obtained by making the sets A and B cliques, but keeping the set of edges between vertices of A and B identical to that of H , i.e. ∀ u ∈ A, v ∈ B , { u, v } ∈ E ( H ∗ ) if and only if { u, v } ∈ E ( H ) . The associated co-bipartite graph H ∗ is not to be confused with the complement of H (i.e. H ) which is also aco-bipartite graph. Definition 3. (Canonical interval representation of a co-bipartite interval graph:) Let I be an AB co-bipartiteinterval graph. A canonical interval representation of I satisfies: ∀ u ∈ A , l ( u ) = l and ∀ u ∈ B r ( u ) = r , wherethe points l and r are the leftmost and rightmost points respectively of the interval representation. We claim that such a representation exists for every AB co-bipartite interval graph. Note that if I is a completegraph, the claim is trivially true. Therefore we take I to be non-complete. Consider any interval representation of I . Since A is a clique there exists a point, say l which is contained in all intervals corresponding to vertices in A .Similarly, let r be a point in the intersection of intervals corresponding to vertices of B . Since I is non-complete,it is clear that l = r . By definition of l and r we have l ( u ) ≤ l ≤ r ( u ) , ∀ u ∈ A and l ( u ) ≤ r ≤ r ( u ) ∀ u ∈ B .Without loss of generality we can assume that l < r and as a result r ( u ) ≥ l and l ( u ) ≤ r for all vertices u . Thismeans no interval ends before the point l and no interval starts after the point r . Hence, it follows that for anyinterval containing l , we can make l its left end point and for an interval containing r , we can make r its rightend point. Therefore, we have a canonical interval representation of I .The following lemma is easy to verify. Lemma 5.
Consider two closed intervals on the real line with left end points l , l and right end points r , r .Then, the two intervals intersect if and only if l ≤ r and l ≤ r . In other words, the two intervals do notintersect if and only if r < l or r < l . Lemma 6.
Let H be an AB bipartite graph and H ∗ its associated co-bipartite graph. If H ∗ is a non-intervalgraph, then box( H ∗ )2 ≤ box( H ) ≤ box( H ∗ ) . If H ∗ is an interval graph, then box( H ) ≤ .Proof. We first show that box( H ) ≤ box( H ∗ ). Let box( H ∗ ) = k ≥ H ∗ = I ∩ I ∩ . . . ∩ I k , where I i areinterval graphs. Note that since I i is a supergraph of a co-bipartite graph, it is a co-bipartite interval graph. Letus consider a canonical interval representation for each I i and further assume that the right end points of allvertices in A and left end points of all vertices in B are distinct. Let I ′ be the interval graph obtained by making r ( u, I ′ ) = l ( u, I ′ ) = l ( u, I ) ∀ u ∈ B and keeping the rest of the intervals unchanged. Similarly, let I ′ be theinterval graph obtained by making l ( u, I ′ ) = r ( u, I ′ ) = r ( u, I ) ∀ u ∈ A . Due to our assumption of distinct endpoints it is clear that A and B are independent sets in I ′ and I ′ respectively. Suppose u ∈ A and v ∈ B . For i ∈ [2]: { u, v } ∈ E ( I ′ i ) ⇐⇒ r ( u, I ′ i ) ≥ l ( v, I ′ i )(by construction of I ′ from I ) ⇐⇒ r ( u, I i ) ≥ l ( v, I i ) ⇐⇒ { u, v } ∈ E ( I i )From this, we immediately see that H = I ′ ∩ I ′ ∩ I ∩ . . . ∩ I k .9ow suppose box( H ∗ ) = 1, i.e. H ∗ is an interval graph. Then we set I = I = H ∗ and proceed as in theprevious case. Hence, box( H ) ≤
2. Note that this inequality is tight: take for example H = C , the cycle of length4. H ∗ is K and therefore an interval graph, but C is not.Now we show that box( H ∗ ) ≤ H ). Let box( H ) = l and H = I ∩ I ∩ . . . ∩ I l , where I i are interval graphs.For each I i , we create two interval graphs I ′ i − and I ′ i as follows: Consider an interval representation of I i . Let l i = min u ∈ V l ( u, I i ) and r i = max u ∈ V r ( u, I i ), the leftmost and rightmost points in the interval representationrespectively. I ′ i − and I ′ i are defined as follows: l ( u, I ′ i − ) = l i and r ( u, I ′ i − ) = r ( u, I i ) , ∀ u ∈ A,r ( u, I ′ i − ) = r i and l ( u, I ′ i − ) = l ( u, I i ) , ∀ u ∈ B,l ( u, I ′ i ) = l i and r ( u, I ′ i ) = r ( u, I i ) , ∀ u ∈ B,r ( u, I ′ i ) = r i and l ( u, I ′ i ) = l ( u, I i ) , ∀ u ∈ A. Now we show that H ∗ = T li =1 I ′ i . From the definitions it is clear that in each I ′ i , A and B are cliques– for example,the interval corresponding to every vertex in A in I ′ i − contains l i . Therefore we will assume that u ∈ A and v ∈ B . { u, v } ∈ E ( H ∗ ) = ⇒ { u, v } ∈ E ( H )= ⇒ { u, v } ∈ E ( I i ) , ∀ i = 1 , , . . . , l (From Lemma 5) = ⇒ l ( u, I i ) ≤ r ( v, I i ) and l ( v, I i ) ≤ r ( u, I i )In I ′ i − , l ( u, I ′ i − ) = l i ≤ r i ≤ r ( v, I ′ i − ) and l ( v, I ′ i − ) = l ( v, I i ) ≤ r ( u, I i ) = r ( u, I ′ i − ) and in I ′ i , l ( v, I ′ i ) = l i ≤ r i ≤ r ( u, I ′ i ) and l ( u, I ′ i ) = l ( u, I i ) ≤ r ( v, I i ) = r ( v, I ′ i ). Therefore u and v are adjacent in both I ′ i − and I ′ i . Now suppose { u, v } / ∈ E ( H ∗ ) = ⇒ { u, v } / ∈ E ( H )= ⇒ ∃ I j such that { u, v } / ∈ E ( I j )In the interval representation of I j , if r ( u, I j ) < l ( v, I j ), then, by definition r ( u, I ′ j − ) < l ( v, I ′ j − ) and hence, { u, v } / ∈ E ( I ′ j − ). If r ( v, I j ) < l ( u, I j ), then, r ( v, I ′ j ) < l ( u, I ′ j ) and therefore, { u, v } / ∈ E ( I ′ j ). Hence proved. ⊓⊔ box( G c ) ≤ box( G ) + 2 : Let box( G ) = k and G = I ∩ I ∩ . . . ∩ I k where I i s are interval graphs. For each I i , weconstruct interval graphs I ′ i with vertex set V ( G c ) as follows: Consider an interval representation for I i . For everyvertex u in I i , we assign the interval of u to u A and u B in I ′ i . Let I ′ k +1 and I ′ k +2 be interval graphs where (1) allvertices in A are adjacent to all the vertices in B (2) In I ′ k +1 A induces a clique and B induces an independentset while in I ′ k +2 it is the other way round. Now we show that G c = I ′ ∩ I ′ ∩ . . . ∩ I ′ k +2 . It is very easy to see that { u A , u B } ∈ E ( I ′ i ) ∀ i ∈ [ k + 2]. Suppose u and v are distinct vertices in G . { u A , v B } ∈ E ( G c ) = ⇒ { u, v } ∈ E ( G )= ⇒ { u, v } ∈ E ( I i ) , i ∈ [ k ]= ⇒ { u A , v B } ∈ E ( I ′ i ) , i ∈ [ k ] . Also, by definition it is clear that { u A , v B } is an edge in both I ′ k +1 and I ′ k +2 . Therefore, I ′ i s are all supergraphsof G c . { u A , v B } / ∈ E ( G c ) = ⇒ { u, v } / ∈ E ( G )= ⇒ ∃ I j , j ∈ [ k ] such that { u, v } / ∈ E ( I j )= ⇒ { u A , v B } / ∈ E ( I ′ j ) . and B induce independent sets in I ′ k +2 and I ′ k +1 respectively. Hence, G c = I ′ ∩ I ′ ∩ · · · ∩ I ′ k ∩ I ′ k +1 ∩ I ′ k +2 andtherefore box( G c ) ≤ box( G ) + 2.box( G ) ≤ G c ) : We will assume without loss of generality that | V ( G ) | >
1. This implies G c is not a completegraph and therefore box( G c ) >
0. Let us consider the associated co-bipartite graph of G c , i.e. G ∗ c . We will showthat box( G ) ≤ box( G ∗ c ) and the required result follows from Lemma 6. Let box( G ∗ c ) = p and G ∗ c = J ∩ J ∩ . . . ∩ J p where J i s are interval graphs. Let us assume canonical interval representation for each J i (recall Definition 3).Corresponding to each J i , we construct an interval graph J ′ i with vertex set V ( G ) as follows: The interval for anyvertex u is the intersection of the intervals of u A and u B , i.e. l ( u, J ′ i ) = l ( u B , J i ) and r ( u, J ′ i ) = r ( u A , J i ). Notethat since u A and u B are adjacent in J i , their intersection is non-empty.Now we show that G = T pi =1 J ′ i . First we consider two adjacent vertices u and v . { u, v } ∈ E ( G ) = ⇒ { u A , v B } , { u B , v A } ∈ E ( G ∗ c )= ⇒ { u A , v B } , { u B , v A } ∈ E ( J i ) , ∀ i ∈ [ p ](From Lemma 5) = ⇒ l ( v B , J i ) ≤ r ( u A , J i ) and l ( u B , J i ) ≤ r ( v A , J i ) , ∀ i ∈ [ p ](By definition of J ′ i ) = ⇒ l ( v, J ′ i ) ≤ r ( u, J ′ i ) and l ( u, J ′ i ) ≤ r ( v, J ′ i ) , ∀ i ∈ [ p ](From Lemma 5) = ⇒ { u, v } ∈ E ( J ′ i ) , ∀ i ∈ [ p ]Therefore, each J ′ i is a supergraph of G . Now, suppose u and v are not adjacent. { u, v } / ∈ E ( G ) = ⇒ { u A , v B } / ∈ E ( G ∗ c )= ⇒ ∃ J j such that { u A , v B } / ∈ E ( J j )(From Lemma 5) = ⇒ r ( u A , J j ) < l ( v B , J j ) or r ( u B , J j ) < l ( v A , J j )(Since J j has a canonical interval representation) = ⇒ r ( u A , J j ) < l ( v B , J j )(By definition of J ′ j ) = ⇒ r ( u, J ′ j ) < l ( v, J ′ j )(From Lemma 5) = ⇒ { u, v } / ∈ E ( J ′ j )Hence, G = J ′ ∩ J ′ ∩ · · · ∩ J ′ p and from Lemma 6 we have box( G ) ≤ box( G ∗ c ) ≤ G c ). References
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