Interval Partitions and Activities for the Greedoid Tutte Polynomial

Transcription

1 Ž. ADVANCES IN APPLIED MAHEMAICS 18, ARICLE NO. AM Interval Partitions and Activities for the Greedoid utte Polynomial Gary Gordon and Elizabeth McMahon Lafayette College, Easton, Pennsylania he two variable greedoid utte polynomial fg; Ž t, z,. which was introduced in previous wor of the authors, is studied via external activities. wo different partitions of the Boolean lattice of subsets are derived and a feasible set expansion of fg Ž. is developed. All three of these results generalize theorems for matroids. One interval partition yields a characterization of antimatroids among the class of all greedoids. As an application, we prove that when G is the directed branching greedoid associated with a rooted digraph D, then the highest power of Ž z 1. which divides fg Ž. equals the minimum number of edges which can be removed from D to produce an acyclic digraph in which all vertices of D are still accessible from the root. he unifying theme behind these results is the idea of a computation tree Academic Press 1. INRODUCION he utte polynomial is an extremely interesting and well studied invariant from graphs and matroids. Many classical counting problems can be expressed in terms of this polynomial, from the chromatic number of a graph to the number of regions into which a family of hyperplanes divides Euclidean space. An extensive introduction to the theory surrounding this invariant can be found in 4. here are three equivalent ways to define the utte polynomial of a matroid: in terms of a deletion-contraction recursion, in terms of a subset expansion Ž the coran-nullity generating function. and via basis activities Ž which was utte s original approach.. In generalizing the utte polynomial from matroids to greedoids in 8, we have taen the subset expansion viewpoint ŽEq. Ž 1.2. below. and found a deletion-contraction recursion ŽEq. Ž 1.3. below.. he two-variable polynomial fg; Ž t, z. Žor simply fg Ž.. which results also has the multiplicative direct sum property: fgg Ž. 1 2 fg Ž. fg Ž his polynomial has been studied for several greedoid classes in 58, $25.00 Copyright 1997 by Academic Press All rights of reproduction in any form reserved.

2 34 GORDON AND MCMAHON In 2, Bjorner, Korte, and Lovasz use the activities approach to define a one-variable greedoid polynomial Ž G.. his approach, for matroids, has close connections with combinatorial topology, especially with the shelling of various simplicial complexes associated with the matroid. Since Ž G. is an evaluation of the two variable polynomial fg, Ž. this leads to the question of how fg Ž. can be viewed in terms of activities in a greedoid. his question is central to this paper. In Section 2, we define a computation tree for a greedoid based on the recursive process of computing fg Ž. using deletion and contraction. his idea is certainly not new; any attempt to compute the utte polynomial recursively will give rise to one. Despite this, we feel that the tree itself is important; it provides the basis for most of the results in this paper. For example, these trees are used in 9 when G is a matroid to unify several different formulations of the utte polynomial. he main results in this section, heorem 2.5 and Proposition 2.7 concern two distinct interval partitions of the Boolean lattice which have direct applications to the polynomial fg. Ž. Both of these results can be viewed as generalizations of a theorem of Crapo for matroids. We also show how a computation tree can be used to give an algorithmic characterization of antimatroids ŽProp- osition In Section 3, we apply 2.5 to fg Ž. to obtain a new feasible set expansion for fg Ž. Ž heorem We also mae explicit the connection between basis activities of 2 and the computation tree. Section 4 is concerned with a direct application of the computation tree idea to directed branching greedoids. We write GD Ž. for the directed branching greedoid associated with a rooted digraph D. he main result Ž heorem 4.2. shows that if Ž z 1. divides fgd;t,z Ž Ž.. with maximum, then is the minimum number of edges which can be removed from D to create a spanning acyclic rooted digraph. We now review some of the mathematical preliminaries we will need. We will assume the reader is familiar with matroids; much of the greedoid terminology is borrowed directly from matroid theory. See Bjorner and Ziegler 3 or Korte, Lovasz, and Schrader 2 for extensive bacground material on greedoids. DEFINIION 1.1. A greedoid G is a pair Ž E, F., where E is a finite set and F is a family of subsets of E satisfying: G1. For every nonempty X F, there is an element x X such that Xx4 F; G2. For X, Y F with X Y, there is an element y Y X such that X y4 F. A subset X F is called feasible. he bases of G are the maximal feasible sets; property G2 ensures that the bases are equicardinal. he

3 HE GREEDOID UE POLYNOMIAL 35 ran of a subset S E, written rž S., is the size of the largest feasible subset of S. A subset S E is spanning if S contains a basis. A loop in a greedoid is an element which is in no feasible set. We occasionally will call these greedoid loops, especially when discussing digraphs, to distinguish them from graph theoretic loops. A coloop is an element which is in every basis of G. Graph theoretic interpretations of these concepts are explored in Section 4. If G is a greedoid on the ground set E, with feasible sets F and ran function r, then we define the two-variable greedoid polynomial fg; Ž t, z. by fž G; t, z. t ržg.ržs. z SrŽs.. Ž 1.2. Ý SE his is the coran-nullity formulation of the utte polynomial when G is a matroid. fg Ž. satisfies the following fundamental deletion-contraction recursion; this underlies all of the recursive structures used throughout this paper. ržg.ržge. If e4 F, then fž G; t, z. fž Ge. t fž Ge.. Ž 1.3. We will need the following characterization of antimatroids. hese characterizations appear in 2, Ž Lemmas 1.2 and PROPOSIION 1.4. Let G Ž E, F. be an accessible set system. hen the following are equialent: Ž a. G is an antimatroid, Ž b. If F, Fx4 and F y4 F, then F x, y4 F, Ž c. If F, F F with F F, then F x4 F implies F x F. 2. COMPUAION REES AND INERVAL PARIIONS If G is a greedoid, a computation tree G Ž. will be a rooted binary tree in which each vertex is labeled by a minor of G in the following way: the two children of a vertex receiving label G will be labeled Ge and Ge, where e4 is feasible in G. DEFINIION 2.1. A computation tree Ž G. for a greedoid G is a rooted labeled tree defined as follows: Ž. a If G has no feasible elements Ži.e., each element of G is a loop., then G Ž. is the trivial labeled rooted tree with a single vertex, labeled by G.

4 36 GORDON AND MCMAHON Ž b. If e4 is feasible in G, then recursively obtain G Ž. by forming the rooted labeled trees for Ge and G e and joining them as in Fig. 1. By applying Ž b. recursively to each leaf of the tree labeled by a greedoid of positive ran, this process will always terminate with the leaves of the computation tree receiving labels which correspond to ran 0 greedoid minors of G Ži.e., each leaf of G Ž. will correspond to a collection of loops.. EXAMPLE 2.2. Let G be the directed branching greedoid associated with the rooted digraph at the top of Fig. 2. Figure 2 then gives a computation tree for G. ŽDirected branching greedoid deletion and contraction correspond to the usual definition of deletion and contraction from graph theory.. We use the convention that the left-child of a vertex is obtained by contraction and the right child comes from deletion. For a computation tree Ž G. with m leaves we introduce the following notation. Let G :1m4 be the set of Ž ran 0. greedoid minors of G which label the leaves of G; Ž. for each, let F represent the elements of G which were contracted in the unique path from the root Žlabeled by G. to the leaf labeled by G, and let ext Ž F. represent the elements of G which correspond to loops in G. Ž We order the labeled leaves from left to right, using the contraction precedes deletion convention of Fig. 2.. hus, ext Ž F. is the set of elements of G which were neither deleted nor contracted along the path in G Ž. from the root to the leaf corresponding to F. We will call x ext Ž F. externally actie for F with respect to G. Ž. For example, ext Ža, c4. b 4, ext Ž. c4 and ext Ža4. for G Ž. in Fig. 2. Different computation trees can give rise to different sets ext Ž F.; for example, if ŽŽ G b. a. c labels a vertex in a different computation tree Ž G., then ext Ža, c4. Žalthough ext Ž. c4 for any tree.. Finally, recall that the closure of S E, denoted by Ž S., is defined by Ž S. x E: rs Ž x4. rž S.4. PROPOSIION 2.3. Let Ž G. be any computation tree for the greedoid G and let F and ext Ž F. be defined as aboe. hen F : 1m4 is precisely the collection of all feasible sets of G and ext Ž F. Ž F. F. FIGURE 1

5 HE GREEDOID UE POLYNOMIAL 37 FIGURE 2 Proof. If G has no feasible singletons, then every element of G is a loop. hen G Ž. is the trivial, single-vertex tree, is the only feasible set and the theorem holds. We may now assume G contains a feasible set e4 and use induction on En. When n 1, G Ž. has two leaves, which are both labeled by the empty greedoid, one corresponding to Ge and the other corresponding to Ge. In this case, we have F e 4, ext Ž F., F, and ext Ž F his tree is unique and clearly satisfies the theorem. Now assume that n 1 and the result holds for all greedoids on n 1 elements. We also assume that e is the first element deleted and contrasted in forming the computation tree, so G Ž. loos as in Fig. 1. Suppose the feasible sets A, A,..., A d of G have been ordered so that e A for 1 j and e A for j 1 d. ŽNote j 1, since e4 is feasible, and jd, since is feasible.. Ge, Ž. by induction, satisfies the theorem, so the leaves of Ge Ž. correspond to the feasible sets of Ge, which by definition are all sets of the form A e 4, where e A and A is feasible in G. hus, F e4 A e 4,so F A for 1 j Ž possibly after reordering these sets.. Similarly, Ge, Ž. by induction, satisfies the theorem, so the leaves of Ge Ž. correspond to feasible sets of G e, which by definition are all feasible sets of G which do not contain e. hus, m d and F A for j 1 m Ž again after possibly reordering these sets.. Since e appears in every feasible set F Ž 1 j. and e

6 38 GORDON AND MCMAHON appears in no feasible set F Ž j1m., these subcomputation trees can be reassembled without repeating any feasible set of G. hus, we have shown that the leaves of G Ž. are in one-to-one correspondence with the feasible sets of G. Finally, ext Ž F. Ž F. F, because if x ext Ž F., then rž F x. rž F., Žor else x4 would be feasible in G, which has ran 0.. hus, x Ž F., but clearly x F. his completes the proof. Antimatroids have been discovered and rediscovered several times. hey were introduced by Dilworth Ž in a lattice-theoretic form. in the 1940s. See 2 or 3 for more bacground on antimatroids. We now use the above theorem to get an algorithmic characterization of antimatroids. PROPOSIION 2.4. Let G be a greedoid and let Ž G. be any computation tree for G. hen ext Ž F. Ž F. F for all feasible sets F if and only if G is an antimatroid. Proof. Let G be an antimatroid, let F be a feasible set of G, and suppose x Ž F. F. We must show x ext Ž F.. Ž he reverse containment ext Ž F. Ž F. F is given by Proposition Let G be the ran 0 greedoid which labels the leaf of G Ž. corresponding to the feasible set F Ž with the correspondence given in Proposition Clearly x was not contracted along the path from the root Ž labeled by G. to G Žsince, by Proposition 2.3, x F if and only if x is contracted along the path.. If x was deleted along this path, let F be the set of elements that were contracted prior to the deletion of x. hen F x4 must be feasible Žsince there is a path in G Ž. in which x is contracted at this stage, and all subsequent elements acted on are deleted.. hen, since G is an antimatroid, we must have F x4 feasible Žby 1.4Ž c... But now x Ž F., which is a contradiction. hus, x was neither deleted nor contracted along the path from the root to the leaf corresponding to F, so xis externally active for F. herefore, ext Ž F. Ž F. F for all feasible sets F. Now suppose ext Ž F. Ž F. F for all feasible sets F. We will show G is an antimatroid by using the local union property which characterizes antimatroids Ž1.4Ž b... Let F, F x4 and F y4 all be feasible sets. We must show F x, y4 is also feasible. If F x, y4 is not feasible, then xžfy4. and y ŽF x 4.. Consider the two paths Px and Py from the root of G Ž. which terminate in the leaves corresponding to Fx4 and F y 4, respectively. Since Px and Py coincide at the root, they must diverge at some vertex of G Ž. which we assume has label G. hen the two children of this vertex must be labeled Gx and G x or they must be labeled Gy and G y, because x and y are the only elements where these two feasible sets differ. Without loss of generality, we may assume the two children are labeled Gx and G x. hen x is

7 HE GREEDOID UE POLYNOMIAL 39 deleted along the path P ; since ext ŽF y4. y is the set of elements which were neither deleted nor contracted, x ext ŽF y4. ŽF y4. ŽFy 4., which contradicts x ŽF y 4.. his completes the proof. We do not consider complexity issues associated with implementing the algorithm suggested by 2.4 here. Instead, we turn our attention to the first main application of Propositions 2.3 and 2.4. he next theorem is a direct generalization of Lemma of 3, which gives a partition of the set S of all spanning sets of a greedoid G into Boolean intervals in which each basis of G is the minimum element of some interval. HEOREM 2.5. Let G Ž E, F. be a greedoid with feasible sets F F, F,...,F 4.hen there is a collection of subsets H Ž F.Ž 1 m. 1 2 m Ž. E such that the interals of the form F, H 1 m partition 2. Furthermore, this partition is unique if and only if G is an antimatroid. In this case, H Ž F. for all. Proof. he sets H which form the maximum elements in the interval E partition of 2 can be obtained directly from a computation tree G: Ž. H F ext Ž F.. he result now follows from 2.3 and 2.4. Since different computation trees can give rise to different external activities, we can also get different interval partitions of 2 E when G is not an antimatroid. For instance, in Example 2.2, the interval a, c 4, a, b, c4 appears in the partition induced by G, Ž. but does not in Ž G.. We now define a notion of internal activity for bases with respect to a computation tree G. Ž. For a basis B of a greedoid G, e B is internally actie with respect to the computation tree G Ž. if e is a feasible coloop in some minor occurring in the unique path from the root to the leaf corresponding to B. he set int Ž B. denotes the set of internally active elements of B for the tree G. Ž. Internal and external activity for matroids are usually defined via a total order of the ground set E. In particular, e B is externally active if e is the least element in the unique Ž basic. circuit contained in B e4 and eb is internally active if e is the least element in the unique Ž basic. bond contained in Ž E B. e 4. Let i Ž B. and e Ž B. denote the set of internally and externally active elements for the basis B with respect to. ŽSee 1 for more details on matroid activity.. he next result shows our definition of internal and external activity for a greedoid based on the computation tree Ž G. coincides with this definition when G is a matroid. We omit the proof. PROPOSIION 2.6. Let G be a matroid with a total order of the ground set E. hen there is a computation tree Ž G. such that i Ž B. int Ž B. and e Ž B. ext Ž B. for all bases B of G.

8 40 GORDON AND MCMAHON he computation tree G Ž. in 2.6 is formed by using to operate on the elements of G in reverse order, starting with the largest nonloop Ž feasible element., so that the sequence of elements deleted and contracted is the same in every path from the root to the leaves. Adhering to a fixed order is always possible for a matroid, although Example 3.3 below shows this is not true for a general greedoid. A theorem of Crapo shows how to get an interval partition of 2 E when G is a matroid such that each interval contains a unique basis of G. ŽSee 1 or 9.. We now generalize this result to greedoids. PROPOSIION 2.7. Let G be a greedoid with a computation tree Ž G. B and bases B, B,...,B. hen the interals of the form B int Ž B. 1 2 m, B Ž. Ž. E ext B 1 m partition 2. Proof. We use induction Ž as in the proof of 2.3.: if rg0, Ž. then the partition is trivial. Let e4 be feasible in G. o simplify notation, we write and B for the bases of Ge and G e, respectively, and let A B int Ž B., Ž. Ž. Ž C B ext B, A B int B, A B int B. e e, C Ž B ext B., and C B ext ŽB.Ž e e where e is the subcompu- tation tree rooted at Ge and e is the subcomputation tree rooted at G e.. If rg0 Ž. and e is a feasible coloop of G, then by induction on Ge, Ž. Ee4 we obtain an interval partition A, C 1 m of 2. hen, for all, B B e 4, int ŽB. int Ž B. e 4, ext ŽB. ext Ž B. e e, 4 E so A A and C C e and this gives the desired partition of 2. Now assume e4 is feasible and is not a coloop. Assume the bases have been ordered so that e B for 1 j and e B for j 1 d Ž where 1 j m since e is not a coloop.. By induction applied to Ge, Ž. Ee4 the intervals of the form A, C 1 j partition 2. Similarly, Ž. Ee4 the intervals of the form A, C j1m also partition 2. In Ge, we find int ŽB. int Ž B. and ext ŽB. ext Ž B. e e ; thus A A e4 and C C e4 Ž for 1 j.. Finally, in G e we again obtain int ŽB. int Ž B. and ext ŽB. ext Ž B. e e,so A A and C C Ž for j 1 m.. his gives the desired partition of 2 E. he two different interval partitions given in 2.5 and 2.7 represent distinct generalizations of Crapo s interval partition for matroids Žheorem of 1 ; see the notes for Section 7.3 there.. hey can also both be viewed as distinct generalizations of Lemma of 3. When G is a E matroid, 2.5 gives a partition of 2 into intervals of the form I, S, where I ranges over all independent sets of G and I S I. For greedoids, the partition of 2.5 is more useful than that of 2.7. he interval partition obtained in 2.5 has the property that every subset in a

9 HE GREEDOID UE POLYNOMIAL 41 given interval has the same ran; this property allows us to prove heorem 3.1. For a general greedoid, the partition of 2.7 is not well-behaved in the same sense. We will explore the difference in more detail in the next section. 3. FEASIBLE SES, ACIVIIES AND HE UE POLYNOMIAL We now apply our computation trees to the polynomial fg. Ž. HEOREM 3.1. Let Ž G. be any computation tree. hen Ý ržg. F extž F. fž G; t, z. t Ž z1.. F feasible Proof. Let FG Ž. denote the family of feasible sets and IŽ F. denote the interval F, F ext Ž F.. he interval partition of heorem 2.5 has the property that every subset S IŽ F. has rž S. F. hen, we can rewrite the subset expansion of 1.2 as Ž. Ý Ý Ý f G; t, z t ržg.ržs. z SrŽS. t ržg.f z SF se FFŽ G. SIŽ F. Ý Ý t ržg.f z SF FFŽ G. SIŽ F. ržg. F extž F. Ý t Ž z1.. FFŽ G. his result gives a way to compute fg Ž. that is more efficient than using all subsets of E as in 1.2. For example, the computation tree Ž G. in Example 2.2 gives fgt Ž. 2 Ž z1. tž z1. 2Ž z1. t1. A general greedoid will still have an exponential number of feasible sets Žas a function of E. ; this is expected in view of results of Jaeger, Welsh, and Vertigan 10 which show almost all evaluations of the utte polynomial are P-complete for many well-behaved classes of matroids. When G is an antimatroid, ext Ž F. Ž F. F for any computation tree G. Ž. In this case, the resulting formula for fg Ž. from 3.1 is given in Proposition 2.2 of 6. We also remar that heorem 3.1 extends Whitney s independent set expansion of the chromatic polynomial of a graph as well as the corresponding expansion of the utte polynomial of a matroid. Furthermore, it provides a direct lin between the basis activities definition of the one-variable greedoid polynomial Ž G. and the two-variable polynomial fg. Ž. We explore this connection in more detail now.

10 42 GORDON AND MCMAHON he usual formulation of Ž G. involves several steps Žsee 2 or 3. : First, choose a total order for the ground set E. Now define an ordering on the bases of G lexicographically so that B1 B2 provided the least lexicographic feasible permutation of B1 precedes the least lexicographic feasible permutation of B 2. Now call x B externally active with respect to the order and the basis B if B B x4 y4 for all y B such that B x4 y4 is a basis of G. ŽIf Bx4y4 is not a basis for any y B, then the condition is vacuously satisfied and x is externally active.. Let ext Ž B. denote the set of all externally active elements. hen define Ž G; u. Ý u ext Ž B.. B basis his notion of external activity for a basis B based on the order Ždeveloped in. 2 can be compared to external activity in the computation tree G. Ž. We now show that these ideas are related by using the order to define a computation tree Ž G.. Given the total order, define Ž G. recursively as before, replacing 2.1Ž b. by: Ž b. If e is the first element in the order with e4 feasible in G, then recursively form Ž G. as in Fig. 1 and remove e from the order. hus, the order is used to decide which element is deleted and contracted at each step. his is similar to the computation tree based on a total order used in Proposition 2.6. PROPOSIION 3.2. Let be a total order of the ground set E of a greedoid G. hen, for all bases B of G, ext Ž B. ext Ž B. for the computation tree Ž G.. Proof. Let e be the first feasible element in and proceed by induction. hen e ext Ž B. for any basis B of G since e is immediately contracted Ž for the bases which contained e. or deleted Žfor the bases which do not contain e. in Ž G.. hus, ext Ž B e. ext Ž B. e for all bases B containing e and ext Ž B. ext Ž B. e for the remaining bases of G. But this relation also holds for ext Ž. : if e is the ordering obtained by removing e from, then ext Ž B e. ext Ž B. e for all bases B containing e and ext Ž B. ext Ž B. e for all bases not containing e. his follows from the proof of heorem 6.4Ž. i of 2. By induction, ext Ž B e. ext Ž B e. for all bases containing e and ext Ž B. e e e ext Ž B. e for all bases not containing e. hus, for the bases containing e, ext Ž B. ext Ž B e. ext Ž B e. ext Ž B. e e, and for the bases not containing e, ext Ž B. ext Ž B. ext Ž B. ext Ž B. e e. his completes the proof.

11 HE GREEDOID UE POLYNOMIAL 43 hus, heorem 3.1 is a direct generalization of the basis activities definition of Ž G.. he order gives a recipe for creating Ž G., but the order may vary for different paths of the computation tree Ž unlie the situation for matroids.. he next example shows how this can occur. EXAMPLE 3.3. Let G be the directed branching greedoid associated with the rooted digraph of Fig. 3. Let B a, c, d, e4 and let B b, c, d, f hen in any computation tree G Ž. associated to G, c must be contracted before d along the path corresponding to the basis B 1, and d must be contracted before c along the path corresponding to the basis B 2. hus, no single total order can be followed along each path in G. Ž. We close this section by interpreting the difference between the two interval partitions given in 2.5 and 2.7 in terms of greedoid polynomials. Let Ž G. be the subtree obtained from a computation tree G Ž. B by simply removing all vertices of G Ž. in which a coloop was deleted. Equivalently, we can view Ž G. B as the subtree formed by all the paths in G Ž. which emanate from the root and terminate at a leaf which corresponds to a basis. hen Ž G. B is the tree which is used to compute the one-variable greedoid polynomial Ž G.. See Fig. 6.3 of 2 for an example. We note that the inductive proof of 2.7 gives a recursive algorithm for creating such a tree Ž G. B. Imitating utte s original approach, it is tempting to create a two-variable greedoid utte polynomial as follows: Let G Ž. be a computation tree for a greedoid G and let BG Ž. denote the family of bases of G. hen define hž G; x, y. Ý x int Ž B. y ext Ž B.. BBG Ž. FIGURE 3

12 44 GORDON AND MCMAHON Unfortunately, internal activity is not well-behaved and this polynomial depends in a crucial way on G. Ž. For example, consider the directed branching greedoid associated with the rooted digraph of Fig. 4. Using a subcomputation tree 1 based on the order a b c d e f gives Ž h1 G x y2x y xy y, while using a subcomputation tree 2 Ž based on the order b a c d e f gives h2g x y x y 2xy 2 y 3. By 3.1, it is easy to see that h Ž G;1, y. fg;0, Ž y1. Ž G; y.. he difference between internal and external activity in greedoids is Ž indirectly. related to the fact that the primal simplicial complex of a greedoid Žwhich is the heredity closure of the family of bases of G. is not shellable in general, while the dual complex Žthe hereditary closure of the family of basis complements. is always shellable. he discussion following heorem of 3 contains more information concerning the relationship between shellability and Ž G.. 4. DIRECED BRANCHING GREEDOIDS: AN APPLICAION When G is a matroid with loops, then Ž z 1. divides fg;t,z; Ž. conversely, when Ž z 1. divides fg;t,z,gmust Ž. have loops. he first implication is true for greedoids Žsee 4.1Ž d. below.; the second is not. We explore this in detail for directed branching greedoids, i.e., the greedoids associated with rooted digraphs. We first list some basic results. If D is rooted digraph with Ž directed. edge set E, the directed branching greedoid GŽ D. Ž E, F. is defined as follows: a set E is feasible in GD Ž. if forms a rooted arborescence of D, i.e., a directed tree in which all of the edges of are directed away from the root. he next result gives graph theoretic interpretations FIGURE 4

13 HE GREEDOID UE POLYNOMIAL 45 for some of the greedoid concepts we have used. See 12 for more discussion. Ž. LEMMA 4.1. Let G D be the directed branching greedoid associated with the rooted digraph D with root ertex *. Ž. a A directed edge e with initial ertex and terminal ertex w is a greedoid loop in GŽ D. if and only if w lies on eery directed path from * to. Ž b. If eery ertex of D is accessible from * ia some directed path, then a subdigraph D will span Ž in the greedoid sense. GŽ D. if and only if, for eery ertex of D, there is a path from * to using only edges of D. ŽSuch a subdigraph D is said to be spanning in D.. Ž. c If D has no greedoid loops and D is spanning in D, then D has no greedoid loops. Ž d. If D has m greedoid loops and D is the digraph obtained from D by remoing these m loops, then fžgž D.. Ž z 1. m fgd Ž Ž... Remars. 4.1Ž. a includes classic loops Žin which the initial and terminal vertices coincide. as well as edges in which the initial vertex is not accessible from *. he condition in 4.1Ž b. about all vertices of D being accessible from * maes sense for the purpose of analyzing fgd Ž Ž... In particular, isolated vertices are irrelevant Žsince the ground set for the greedoid is the edge set.; thus we always assume D has no isolated vertices. Further, if D has an inaccessible vertex Ž from *., any edge incident to will be a greedoid loop Žas 4.1Ž a. is vacuously satisfied.. Since greedoid loops behave in a predictable way with respect to the utte polynomial Ž4.1Ž d.., we will concentrate on digraphs D in which there are no isolated vertices or greedoid loops. We now turn our attention to the main results of this section. When G is the directed branching greedoid associated with a rooted digraph, we obtain Ž heorem 4.2. a graph theoretic interpretation for the highest power of Ž z 1. which divides fg. Ž. his result generalizes heorem 3 of 12, where it was shown that when G is the directed branching greedoid associated with a rooted digraph D having no greedoid loops, then D has a directed cycle if and only if Ž z 1. divides fg. Ž. HEOREM 4.2. Let G Ž GŽ D.. be the directed branching greedoid associated with a rooted digraph D with no greedoid loops or isolated ertices. If fž G; t, z. Ž z 1. f Ž t, z., where Ž z 1. does not diide f Ž t, z. 1 1, then is the minimum number of edges that need to be remoed from D to leae a spanning, acyclic directed graph. Proof. Assume D is a rooted directed graph with no greedoid loops, and fgd;t,z Ž Ž.. Ž z1. f Ž t, z., where Ž z 1. does not divide f Ž t, z.. 1 1

14 46 GORDON AND MCMAHON Let m be the minimum number of edges whose removal leaves an acyclic, spanning subdigraph. Part 1. m. We will construct a set of edges in D whose removal leaves an acyclic, spanning digraph. Let G Ž. be a computation tree for G and let e be the first edge deleted and contracted in G. Ž. hen the multiplicity of Ž z 1. in fg;t,z Ž. is exactly equal to the smaller multiplicity of Ž z 1. in fge;t,z Ž. or fge;t,z Ž. ŽLemma 1 of 12.. Applying this repeatedly, we can form a path P in G Ž. by choosing either He or H e for every subgreedoid H encountered along the way, according to which of fž He. or fž He. contains exactly factors of Ž z 1.If. both fž He. and fž He. contain exactly factors of Ž z 1., choose He to form the paths. hus, our path P in G Ž. begins at the root, ends at a leaf, and has the property that Ž z 1. has multiplicity in fž H. for every subgreedoid H on P. Let B be the feasible set which labels the leaf of P. hen by heorem 3.1, ext Ž B.. Now let D be the rooted digraph obtained from D after the edges corresponding to these elements of G are removed from D Ži.e., after ext Ž B. is removed from G. and let G Ž GD Ž.. be the directed branching greedoid associated with D. We form a computation tree G Ž. which has the property that the sequence of deletions and contractions along one path P will be precisely the same as the sequence along P in G Ž. Žthe rest of G Ž. can be formed arbitrarily.. he minor which labels the leaf of the path P is empty Ži.e., the corresponding feasible set of G has no externally active elements., since every edge was either deleted or contracted. hus, the polynomial for that minor has no factors of Ž z 1., so the polynomial fg Ž. has no factors of Ž z 1. by heorem 3.1. By Lemma 4.1Ž d., D has no greedoid loops. hus, we can apply heorem 3 of 12 to conclude that D is acyclic. It remains to show that D spans D. Suppose that this is not the case. hen there is a vertex which is not reachable from * in D. Since is reachable from * in D, there must be consecutive nodes labeled H and He Ž for some edge e. along the path P in the computation tree such that there is a path from * to in the digraph corresponding to H, but no such path in the digraph corresponding to H e. ŽNote that the edge e must be deleted along P.. Let w be the terminal vertex of e and let e be an edge in a path from * to Ž in the digraph corresponding to H. whose initial vertex is w. hen the set B e4 is feasible and e ext Ž B.. Without loss of generality, we may assume that the computation tree G Ž. contains a path P which agrees with the path P from the root to the node labeled by the minor H, then selects the node labeled by He, and then uses precisely the same sequence of deletions and contractions which occurred along the path P. Further, the path P must give rise to a

15 HE GREEDOID UE POLYNOMIAL 47 higher number of externally active elements at its node, by the way the path P was chosen. o see why it is always possible to find such a path P, we must show that if f is feasible in a minor H d of H e, then f is also feasible in the corresponding minor H c of He. But this is clear, since there must be a path from * which includes the edge f in H e and this path avoids the vertex w Žas e ext Ž B... hus the same path including f exists in He, and f will be feasible in H c. Since * and w coincide in He, the edge e must eventually be deleted or contracted in the computation tree. hus e ext Ž B e 4.. We claim that ext Ž B e4., producing a term t a Ž z1. b in fg;t,z Ž. with b Ž by heorem 3.1., which would contradict the minimality of. o show this, we will prove that ext ŽB e4. ext Ž B.. Suppose d ext ŽB e4. and let HŽ d. denote the minor labeling a node on the path P in which d first becomes a greedoid loop. here are three possibilities for where HŽ d. can appear along P. Case 1. HŽ d. labels a node which P shares with P. Clearly d will be a greedoid loop in all of the descendants of HŽ d.,so dext Ž B.. Case 2. HŽ d. He. hen either d and e are parallel edges or d and e must share the same terminal vertex w, from 12. But d cannot be parallel to e in the digraph corresponding to H, since there is no path from * to in the digraph corresponding to H e. Further, since is not reachable from * in the digraph corresponding to H e, any edge d e having terminal vertex w will be a greedoid loop in H e, i.e., d ext Ž B.. Case 3. HŽ d. is a proper minor He. If dis not a greedoid loop in He, then by 4.1Ž. a there is at least one feasible path from * to the terminal vertex of d in H e. Furthermore, as before, no such path can pass through the vertex w. hus each such path in H e gives rise to a corresponding path in He. he sequence of deletions and contractions which produce HŽ d. must destroy all these paths in He; hence the same sequence of deletions and contractions will also destroy all such paths in a corresponding minor of H e, i.e., d ext Ž B.. We have shown ext Ž B e4. ext Ž B. e 4, so ext ŽB e4.. his gives us the desired contradiction. hus we have produced edges whose removal leaves an acyclic, spanning subgraph. his completes this part of the proof. Part 2. m. We reverse the process used in the first part of the proof. First choose a minimal set of m edges whose removal leaves an acyclic, spanning digraph D. Let G Ž. be any computation tree for the directed branching greedoid G Ž GD Ž... hen, by heorem 3 of 12, Ž z1. does not divide fg, Ž. so, by heorem 3.1, there is a path P in G Ž. from the root to a leaf labeled by an empty minor, i.e., a feasible set with no external activity. We now create a computation tree for GD. Ž.

16 48 GORDON AND MCMAHON FIGURE 5 Consider the sequence of deletions and contractions along P in G Ž. and form a computation tree G Ž. with a path P that has precisely the same sequence of deletions and contractions as P does Žwith the rest of G Ž. completed arbitrarily.. ŽAgain, this is always possible, since adding the m edges to D will not mae an edge which was feasible in D become not feasible in D.. Suppose this process terminates at a vertex of G Ž. labeled by the minor H. By heorem 3.1 and the construction of the path P, either H or one of its descendants will contribute the term t a Ž z1. b to the polynomial fg Ž. for some b m. But b since Ž z 1. divides each term of fg,som. Ž. his completes the proof. heorem 4.2 generalizes a similar result which holds for the polynomial Ž G.. See the discussion concerning directed branching greedoids following Proposition in 3. he necessity of the spanning condition in 4.2 can be seen in the following example. EXAMPLE 4.3. Let D be the rooted digraph of Fig. 5. hen Ž z 1. 2 divides the utte polynomial, so the minimum number of edges which can be removed to leave an acyclic spanning digraph is 2. Removing the edge labeled a leaves an acyclic digraph, but the resulting digraph does not span. ACKNOWLEDGMEN We wish to than the anonymous referee for many helpful suggestions in clarifying the exposition in Section 4 and the proof of heorem 4.2 and for contributing Example 4.3.

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