Decompositions and Forcing Relations in Graphs and Other
4.2. The Delta Tree and Interval Orientations
In this section, we show that the Delta tree has a role in interval orientations that is analogous to the role of modular decomposition in transitive orientations.
The following is analogous to Lemma 3.17.
Lemma 4.7 If M is a Delta module of T ,(a,b)is an edge such that{a,b} ⊆M and (c,d)is an edge such that{c,d} ⊆ M, then(a,b)and(c,d)are in different components of I(G).
Proof.Suppose that they are in the same component. There is a path inI(G) from (a,b) to (c,d). There must therefore be an adjacent pair (u, v),(v, w) inI(G) such that{u, v} ⊆ Mand{v, w} ⊆M.
Suppose (u, v) and (v, w) fall into case (b) or case (c) of Definition 4.2. Then the relationship ofwtoudiffers from its relationship tov. Sinceu, v∈M andwis not, M cannot be a module, a contradiction.
Otherwise, (u, v) and (v, w) fall into case (d) of Definition 4.2. Ifuvis an edge of G1anduwis an edge ofGn, thenwvis an edge ofG1, henceM fails to be a module inG1, a contradiction. Thus,uvis an edge of Gn anduw andvware edges of G1. However, this means thatM is not a clique, and it has outgoing edges ofG1tow.M again fails to be a Delta module, a contradiction. Q.E.D.
Let us say thatT isprimeif it has no modules, andDelta-primeif it has no Delta modules. IfT is prime, then it is also Delta prime, but the converse is not true.
Let us now redefine the pivot operation of Algorithm 3.20 so that we can use the algorithm to find an interval orientation, rather than a transitive orientation. As before, letP =(X1,X2, . . . ,Xk) be a partition of the vertices ofG1ninto more than one partition class, together with a linear ordering of members ofP. Apivotis the following operation. Select a non-singleton class Xi. In this case, a pivot vertex is a vertexz∈V −Xi such that the relationship ofzto members of Xi is non-uniform.
That is, in at least one ofGc,G1, andGn,zhas both neighbors and non-neighbors inXi.
Let j be the index of the partition classXj that containsz. LetYc,Y1, andYnbe the members of Xi that are neighbors of zinGc,G1, andGn, respectively. If j >i, then refine the partition by changing the partition and ordering to (X1,X2, . . . ,Xi−1, Yn,Y1,Yc, Xj+1, . . . ,Xk). Otherwise change it to (X1, X2, . . . , Xi−1, Yc,Y1, Yn, Xj+1, . . . ,Xk). At least two of the sets Xc, X1, andXn are nonempty because of the choice ofz. If one of the sets is empty, remove it from the partition. Let us call this type of pivot athree-way pivot.
There is one other type of pivot that must be used whenXiis a module that fails to be a Delta module. This is amodular pivot, which consists of the following operation.
Letzbe a vertex inV −Xisuch that the vertices inXiare neighbors inG1. SinceXi
is not a Delta module, such azexists andXi is not a clique. Letxbe a vertex with an incident edge inGn|Xi, and letY be the neighbors ofxinGn|Xi. We replaceXiin the ordering withXi−Y andY, placingXi−Y first ifzresides in a class that is later in the ordering thanXi, and placing it second ifzresides in an earlier class.
The following observation is the analog of Observation 3.19 in the context of interval orientations.
Observation 4.8 Let Yn, Y1, and Yc be as in the description of a three-way pivot. If x∈Yn, then all edges of G1nfrom x to Y1∪Ycare neighbors of(z,x)in I(T), hence even-equivalent to(x,z). Similarly, if y∈Y1, then all edges of G1n from y to Ycare even-equivalent to(y,z). In particular, if all edges of G1n in Yn∪Yc× {z}are even-equivalent, then so are all edges of G1n that are directed from earlier to later sets in the sequence(Yn,Y1,Yc).
The following gives a similar insight about the effect of a modular pivot:
Lemma 4.9 Let Xi, Y , x, and z be as in the definition of a modular pivot. All edges of G1n that are directed from Xi−Y to Y are in the same even-equivalence class as (x,z)in I(T).
Proof.All members of{x} ×Yare directed edges ofGn, and sincezhas edges ofG1to all of{x} ∪Y, it is immediate that these edges are neighbors of (z,x) inI(T), hence even-equivalent with (x,z).
Letabbe an arbitrary edge ofG1nsuch thata ∈Xi−Y,a =x, andb∈Y. Note thataxis not an edge ofGn, by the definition ofY.
If ab is an edge of Gn, then ab, xb are both edges in Gn and ax is not an edge ofGn. Thus, (a,b) is even-equivalent to (x,b), and transitively, even-equivalent to (x,z).
Supposeabis an edge of G1. Then ax is either an edge inG1 or inGc, but in either of these cases, (a,b) and (b,x) are neighbors in I(T), hence (a,b) and (x,b) are even-equivalent. Transitively, (a,b) is even-equivalent to (x,z). Q.E.D.
The following is analogous to Lemma 3.21.
Lemma 4.10 Let T be Delta-prime, and let(v1, v2, . . . , vn)be an ordering of the vertices returned by the variant of Algorithm 3.20, that uses three-way and modular pivots in place of the standard pivot. Then the edges of G1n that are directed from earlier to later vertices in this ordering are even-equivalent in I(G).
The proof differs only in trivial details from the proof of Lemma 3.21. The main difference is that it uses Observation 4.8 and Lemma 4.9 in place of Observation 3.19 at each place in the proof. The results of the section on comparability graphs are now easily adapted to the new context; the details of these proofs are also trivial and are left to the reader:
Theorem 4.11 [23] Let T be an arbitrary intersection matrix.
1. If X and Y are two children of a degenerate node in the Delta tree that are adjacent in G1n, then(X×Y)∪(Y×X)is a connected component of I(G).
If T is an interval matrix, then{X×Y,Y×X}is its bipartition.
2. If X1,X2, . . . ,Xkare children of a prime node, then the set EZ = {(a,b) :ab is an edge of G1nand x∈ Xi,y∈ Xj,i= j}is a connected component of I(G).
If T is an interval matrix, then there exists an ordering Xπ(1),Xπ(2), . . . ,Xπ(k)
such that the elements of EZthat go from an earlier child to a later child are one bipartition class and those that go from a later child to an earlier are another.
3. There are no other connected components of I(T).
Corollary 4.12 An intersection matrix is an interval matrix iff I(G)is bipartite.
Lemma 4.13 If T is an n×n intersection matrix that is not an interval matrix, then I(T)has an odd cycle of length at most3n.
Recall that we can represent an interval realizer with a string, where each letter of the string is the name of a vertex whose interval has an endpoint there. SupposeT is an interval matrix. A strong Delta modulesMofThas an analog in an interval realizerR ofT:Mis a set of intervals whose left endpoints are a consecutive substring and whose right endpoints are a consecutive substring. If M has a neighbor y∈V−M inG1, then intervalymust have an endpoint inside all intervals inMand an endpoint outside all intervals inM. This constrains the intervals inM to have a common intersection, which forcesGn|M to have no edges.
The weak modules are those that satisfy these requirements in some, but not all, realizers ofT. The correspondence with Delta modules ofTmeans that such sets define a quotient structure on the class of all interval realizers.
When one labels the nodes of the Delta tree with interval representations of their as-sociated quotients, the tree gives a representation of an interval realizer (see Figure 15).
This is obtained by a composition operation on the interval realizers themselves. This reflects the fact that the Delta modules define a quotient structure on interval realizers that mirror the quotient structure that they define on the intersection matrices of the interval realizers.
The quotient at a prime node can be reversed to obtain a representation of another quotient. A degenerate node induces a quotient that is complete in one ofGn,G1, orGc; it is easy to see how a realizer of such a graph can be permuted without changing such a simple represented quotient inT. Since each realizer ofTis a string, the collection of all realizers ofT is a language, and this tree gives a type of grammar for that language.
An algorithm is given in [22] for finding the modular decomposition of a graph that usesI(G) to reduce the problem to theRefineoperation of Algorithm 3.20. The similarities between the modular decomposition and itsrelation, on the one hand, and the Delta tree and its Delta relation, on the other, can be used to reduce finding
l m
Figure 15. An interval realizer and the Delta tree for the intersection matrix given by a realizerR. When Mis an internal node andCis its children, the node is labeled with the quotient (R|M)/C, represented here with their string representations. By performing substitution operations in postorder, it is possible to reconstruct Rin O(n) time using a composition operation that mirrors the one defined by the quotient structure.
the Delta tree to the variant ofRefineoperation that uses the three-way and modular pivots in the same way. This, together with a linear-time algorithm forRefinethat is given in [21] is the basis of an algorithm given in [17, 23] for finding the Delta tree in time linear in the size of the intersection graphG1cthat it represents. This assumes a sparse representation of the matrix consisting of labeling the edges ofG=G1cwith with 1’s andc’s.