Duality Respecting Representations and Compatible Complexity

Duality Respecting Representations and Compatible Complexity Measures for Gammoids Immanuel Albrecht FernUniversität in Hagen, Department of Mathematics and Computer Science, Discrete Mathematics and Optimization D-58084 Hagen, Germany Abstract We show that every gammoid has special digraph representations, such that a representation of the dual of the gammoid may be easily obtained by reversing all arcs. In an informal sense, the duality notion of a poset applied to the digraph of a special representation of a gammoid commutes with the operation of forming the dual of that gammoid. We use these special representations in order to define a complexity measure for gammoids, such that the classes of gammoids with bounded complexity are closed under duality, minors, and direct sums. Keywords: gammoids, digraphs, duality, complexity measure 2010 MSC: 05B35, 05C20, 06D50 A well-known result due to J.H. Mason is that the class of gammoids is closed under duality, minors, and direct sums [1]. Furthermore, it has been shown by D. Mayhew that every gammoid is also a minor of an excluded minor for the class of gammoids [2], which indicates that handling the class of all gammoids may get very involved. In this work, we introduce a notion of complexity for gammoids which may be used to define subclasses of gammoids with bounded complexity, that still have the desirable property of being closed under duality, minors, and direct sums; yet their representations have a more limited number arXiv:1807.00588v2 [math.CO] 27 Jun 2020 of arcs than the general class of gammoids. 1. Preliminaries In this work, we consider matroids to be pairs M = (E, I) where E is a finite set and I is a system of independent subsets of E subject to the usual axioms ([3], Sec. 1.1). If M = (E, I) is a matroid and X ⊆ E, then the restriction of M to X shall be denoted by M|X ([3], Sec. 1.3), and the contraction of M to X shall Email address: [email protected] (Immanuel Albrecht) Preprint submitted to Discrete Mathematics June 30, 2020 be denoted by M.X ([3], Sec. 3.1). Furthermore, the notion of a digraph shall be synonymous with what is described more precisely as finite simple directed graph that may have some loops, i.e. a digraph is a pair D = (V, A) where V is a finite set and A ⊆ V × V . Every digraph D = (V, A) has a unique opposite digraph Dopp = (V, Aopp) where (u, v) ∈ Aopp if and only if (v,u) ∈ A. All standard notions related to digraphs in this work are in accordance with the definitions found in [4]. A path in D = (V, A) is a non-empty and non-repeating sequence p = (p1p2 ...pn) of vertices pi ∈ V such that for each 1 ≤ i<n, (pi,pi+1) ∈ A. By convention, we shall denote pn by p−1. Furthermore, the set of vertices traversed by a path p shall be denoted by |p| = {p1,p2,...,pn} and the set of all paths in D shall be denoted by P(D). Definition 1.1. Let D = (V, A) be a digraph, and X, Y ⊆ V . A routing from X to Y in D is a family of paths R ⊆ P(D) such that (i) for each x ∈ X there is some p ∈ R with p1 = x, (ii) for all p ∈ R the end vertex p−1 ∈ Y , and (iii) for all p, q ∈ R, either p = q or |p| ∩ |q| = ∅. We shall write R: X → Y in D as a shorthand for “R is a routing from X to Y in D”, and if no confusion is possible, we just write X → Y instead of R and R: X → Y . A routing R is called linking from X to Y , if it is a routing onto Y , i.e. whenever Y = {p−1 | p ∈ R}. Definition 1.2. Let D = (V, A) be a digraph, E ⊆ V , and T ⊆ V . The gammoid represented by (D,T,E) is defined to be the matroid Γ(D,T,E) = (E, I) where I = {X ⊆ E | there is a routing X → T in D}. The elements of T are usually called sinks in this context, although they are not required to be actual sinks of the digraph D. To avoid confusion, we shall call the elements of T targets in this work. A matroid M ′ = (E′, I′) is called gammoid, if there is a digraph D′ = (V ′, A′) and a set T ′ ⊆ V ′ such that M ′ = Γ(D′,T ′, E′). Theorem 1.3 ([1], Corollary 4.1.2). Let M = (E, I) be a gammoid and B ⊆ E a base of M. Then there is a digraph D = (V, A) such that M = Γ(D,B,E). For a proof, see J.H. Mason’s seminal paper On a Class of Matroids Arising From Paths in Graphs [1]. Here, we are content with pointing out that the proof is constructive and involves a sequence of the following kind of construction: Definition 1.4. Let D = (V, A) be a digraph, and let (r, s) ∈ A be an arc of D. The swap of (r, s) in D shall be the digraph D(r,s) = (V, A(r,s)) where A(r,s) = {(u, v) ∈ A | u 6= r}∪{(s, v) | v 6= s and (r, v) ∈ A}∪{(s, r)}. 2 First, observe that the number of arcs in the swap of (r, s) in D is bounded by the number of arcs in the original digraph D. The proof of Theorem 1.3 in [1] is based on the observation that the gammoid Γ(D,T,E) is the same as the gammoid Γ(D(r,s),T(r,s), E) where D = (V, A), T(r,s) = (T \{s}) ∪{r}, and (r, s) ∈ A such that s ∈ T and r∈ / T . Remark 1.5. For every representation (D,T,E) and every routing R: B → T , there is a straight-forward sequence of swaps that yields a representation of the desired form: assume that R = r(1),...,r(n) thus fixing a particular order of the paths in R. Each path in the routing corresponds to a con- tiguous subsequence of swaps which appear in the above order. The path (i) (i) (i) (i) r = r1 r2 ...rni corresponds to a block of ni − 1 swaps, indexed by (i) k = 1, 2,...,ni − 1. The kth swap in the subsequence corresponding to r (i) (i) (k+Ki−1) is the swap of rni−k, rni−k+1 in the digraph D , which yields the digraph (k+Ki) (k+Ki−1) D = D i i r( ) ,r( ) ni−k ni−k+1 j<i (0) where Ki = j=1(nj − 1) and D = D. In other words,P given a basis B of Γ(D,T,E), we may swap all arcs of a given routing B → T in D in the reverse order of traversal in their respective paths, which gives us a digraph D′ such that Γ(D,T,E)=Γ(D′,B,E). Furthermore, all b ∈ B\T will be sinks in D′, and for the sake of reduction of complexity, we remove all arcs that start in b ∈ B ∩ T from D′, too; this does not change the represented gammoid. 2. Special Representations Definition 2.1. Let (D,T,E) be a representation of a gammoid. We say that (D,T,E) is a duality respecting representation, if Γ(Dopp, E\T, E) = (Γ(D,T,E))∗ where (Γ(D,T,E))∗ denotes the dual matroid of Γ(D,T,E). Lemma 2.2. Let (D,T,E) be a representation of a gammoid with T ⊆ E, such that every e ∈ E\T is a source of D, and every t ∈ T is a sink of D. Then (D,T,E) is a duality respecting representation. Proof. We have to show that the bases of N = Γ(Dopp, E\T, E) are precisely the complements of the bases of M = Γ(D,T,E) ([3], Thm. 2.1.1). Let B ⊆ E be a base of M, then there is a linking L: B → T in D, and since T consists of sinks, we have that the single vertex paths {(x) ∈ P(D) | x ∈ T ∩ B} ⊆ L. opp opp Further, let L = {(pnpn−1 ...p1) | (p1p2 ...pn) ∈ L}. Then L is a linking from T to B in Dopp which routes T \B to B\T . The special property of D, 3 that E\T consists of sources and that T consists of sinks, implies, that for all p ∈ L, we have |p| ∩ E = {p1,p−1}. Observe that thus opp opp R = {p ∈ L | p1 ∈ T \B}∪{(x) ∈ P(D ) | x ∈ E\ (T ∪ B)} is a linking from E\B = (T ∪˙ (E\T )) \B onto E\T in Dopp, thus E\B is a base of N. An analog argument yields that for every base B′ of N, the set E\B′ is a base of M. Therefore Γ(Dopp, E\T, E) = (Γ(D,T,E))∗. Definition 2.3. Let M be a gammoid and (D,T,E) with D = (V, A) be a representation of M. Then (D,T,E) is a standard representation of M, if (i) T ⊆ E, (ii) every t ∈ T is a sink in D, and (iii) every e ∈ E\T is a source in D. The name standard representation is justified, since the real matrix representation A ∈ RT ×E of M obtained from D through the Lindström Lemma [5, 6] is a standard matrix representation of Γ(D,T,E) up to possibly rearranging the columns ([7], p.137).

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