Comparability Invariance Results for Tolerance Orders Kenneth P. Bogart Garth Isaak∗ Department of Mathematics Department of Mathematics Dartmouth College Lehigh University Hanover, NH 03755 Lehigh, PA 18015 Joshua D. Laison Ann N. Trenk† Department of Mathematics Department of Mathematics Dartmouth College Wellesley College Hanover, NH 03755 Wellesley, MA 02481 March 15, 2001 ABSTRACT We prove comparability invariance results for three classes of ordered sets: bounded tolerance orders (equivalent to parallel- ogram orders), unit bitolerance orders (equivalent to point-core bitolerance orders) and unit tolerance orders (equivalent to 50% tolerance orders). Each proof uses a different technique and relies on the alternate characterization. 1 Background Throughout this paper we consider only finite ordered sets. We begin with background material on comparability invariance. A property or parameter ∗Research supported in part by the Reidler Foundation. †Research supported in part by the American Association of University Women and Cornell University. 1 of an ordered set is said to be comparability invariant if all orders with a given comparability graph have that property or have the same value of that parameter. The first printed reference we know of to the phrase “comparability invari- ant” is in Michel Habib’s paper [11]. However Habib [Personal Communica- tion, December 2000] tells us that he first learned the phrase in conversations with J.C. Arditti and M.C. Golumbic. The interest in comparability invari- ants arises from papers of Arditti [1], Gysin [10], and Trotter, Moore, and Sumner [17] in which they show that all transitive orientations of a finite comparability graph have the same dimension, and one of Arditti and Jung [2] in which they extend the result to infinite comparability graphs. Trotter, Moore, and Sumner attribute the question of whether all orders having the same comparability graph have the same dimension to Bogart. The paper [12], by Habib, Kelly, and M¨ohring has brought the idea of a comparability invariant to the attention of a broader audience and is one of the motivations for the current paper. A second motivation is a conversation between one of the authors (Trenk) and Habib in 1999. It is natural to ask whether any “reasonable” property of an ordered set is a comparability invariant. The ordered set with elements t, b,anda1, a2, and a3,withb ≺ ai ≺ t and no other relations has a comparability graph isomorphic with the ordered set t1, t2, t3, a,andb,withb ≺ a ≺ ti and no other relations. However the number of maximal elements of the first ordered set is one, as is the number of minimal elements, while the number of maximal elements of the second ordered set is three, while the number of minimal elements is one. Thus neither the number of maximal elements nor the total number of maximal and minimal elements is a comparability invariant. In this paper we show that the property of belonging to set S is a compara- bility invariant for S = {bounded tolerance orders}, S = {unit bitolerance orders}, and S = {unit tolerance orders}. Each of these classes has an alternate characterization: bounded tolerance orders are equivalent to parallelogram orders, unit bitolerance orders are equivalent to point-core bitolerance orders, and unit tolerance orders are equivalent to 50% tolerance orders. We review these alternate characterizations which will be used in the comparability in- variance proofs in section 2. 2 1.1 Comparability Invariance We next present the standard technique for proving that an ordered set property is a comparability invariant. Given a graph G =(V,E), a set A ⊆ V is called an autonomous set if every vertex in V \ A is either adjacent to all of the vertices in A or none of the vertices in A. Autonomous sets play a key role in relating ordered sets that have the same comparability graph. Let P =(V,≺1)andQ =(V,≺2) be ordered sets with the same compara- bility graph G.WesaythatQ is obtained from P by an elementary reversal if there is a set A ⊆ V that is autonomous in G and satisfies the following. 1. A is not an independent set of G. 2. If x, y are not both in A then x ≺1 y iff x ≺2 y. 3. If x, y ∈ A then x ≺1 y iff y ≺2 x. In this process, Q is obtained from P (and vice versa) by reversing the comparabilities in A. Note that the definition of an autonomous set A in the comparability graph of an order P allows for the possibility of an element x which is above some elements of A and below others. However, the second and third conditions above imply that this is not possible in an autonomous set that participates in an elementary reversal. We record this below in a remark. In this paper we will only be concerned with autonomous sets that participate in elementary reversals, called order autonomous sets in [14]. Remark 1 If Q =(V,≺2) is obtained from P =(V,≺1) by an elementary reversal using the order autonomous set A, then the sets Pred(A)={v ∈ V | x ≺1 a for all a ∈ A}, Succ(A)={w ∈ V | a ≺1 w for all a ∈ A},and Inc(A)={z ∈ V | z a for all a ∈ A} partition V \ A. By the second condition in the definition of an elementary reversal, we could also use the relation ≺2 of Q in defining the sets Pred(A), Succ(A), and Inc(A). If A is an order autonomous set and a ∈ A is incomparable to every other element of A,thenA = A \{a} is another order autonomous set. Furthermore, Q can be obtained from P by an elementary reversal of A if and only if Q can be obtained from P by an elementary reversal of A.When all such elements are removed from an order autonomous set, the resulting set will not be empty by the first condition of our definition. We record this as a remark. 3 Remark 2 If one ordered set is obtained from another by an elementary reversal, this can be achieved using an order autonomous set A in which every element of A is comparable to another element of A. The following theorem of Gallai [7] (which appears in [16], pgs. 61–62) shows that we can move between any two orders with the same comparability graph by a sequence of elementary reversals. Theorem 3 (Gallai) Let G =(V,E) be the comparability graph associated with distinct ordered sets P =(V,≺P ) and Q =(V,≺Q). Then there exists a sequence of ordered sets P0,P1,...,Pm so that P0 = P , Pm = Q and Pi+1 is obtained from Pi by an elementary reversal for i =0, 1,...,m− 1. Theorem 3 allows us to show a property is comparability invariant by considering pairs of orders for which one can be obtained from the other by an elementary reversal. A corollary of Theorem 3 which will be useful to us is given below. Corollary 4 Let P and Q be finite ordered sets with the same comparability graph and let S be a class of orders. To prove that P ∈S ⇐⇒ Q ∈S, it suffices to prove P ∈S⇒Q ∈Swhere Q can be obtained from P by an elementary reversal. Proof. By Theorem 3 we need only prove P ∈S⇐⇒Q ∈Sin the case that Q can be obtained from P by an elementary reversal. However, since the process of obtaining one order from another by an elementary reversal is symmetric, the result follows. ¾ 1.2 Classes of Bounded Bitolerance Orders A bounded bitolerance representation I,p,q of an order P =(V,≺) consists of a function I that maps each element v ∈ V to an interval Iv =[L(v),R(v)] on the real line and a pair of functions p and q that map each element v of V to points p(v)andq(v) in the interval Iv with p(v) = L(v)andq(v) = R(v). The point p(v) is called the left tolerant point of v and q(v) is called the right tolerant point of v. In a bounded bitolerance representation of P =(V,≺), we have x ≺ y if and only if R(x) <p(y)andq(x) <L(y). An ordered set is a bounded bitolerance order if it has such a representation. Given a bounded bitolerance representation, the left tolerance of element v is the 4 quantity tl(v)=p(v) − L(v), and the right tolerance of element v is the quantity tr(v)=R(v) − q(v). Bounded bitolerance orders are the orders of interval dimension two, as first observed in [15]. In [12] the property of having interval dimension 2 was shown to be a comparability invariant. Another way to represent a bounded bitolerance order uses trapezoids. Let L1 and L2 be horizontal lines with L1 above L2. We consider trapezoids Tv that have one base on L1 and the other base on L2, and we allow degenerate trapezoids in which either or both bases is a point. For trapezoids Tv with bases on L1 and L2, we write Tx Ty when Tx ∩ Ty = ∅ and every point of Tx is to the left of some point of Ty, which we shorten to “Tx is to the left of Ty.” Similarly for intervals Ix,Iy we will write Ix Iy when Ix ∩ Iy = ∅ and Ix is to the left of Iy. A trapezoid representation of P =(V,≺) is a function T that assigns to each v ∈ V a trapezoid Tv with one base on L1 and the other base on L2, so that x ≺ y iff Tx Ty.