Gammoid

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In matroid theory, a field within mathematics, a gammoid is a certain kind of matroid, describing sets of vertices that can be reached by vertex-disjoint paths in a directed graph.

The concept of a gammoid was introduced and shown to be a matroid by Hazel Perfect (1968), based on considerations related to Menger's theorem characterizing the obstacles to the existence of systems of disjoint paths.[1] Gammoids were studied in more detail and given their name by Mason (1972).[2]

Definition[edit]

Let be a directed graph, be a set of starting vertices, and be a set of destination vertices (not necessarily disjoint from ). The gammoid derived from this data has as its set of elements. A subset of is independent in if there exists a set of vertex-disjoint paths whose starting points all belong to and whose ending points are exactly .[3]

A strict gammoid is a gammoid in which the set of destination vertices consists of every vertex in . Thus, a gammoid is a restriction of a strict gammoid, to a subset of its elements.[3][4]

Example[edit]

Consider the uniform matroid on a set of elements, in which every set of or fewer elements is independent. One way to represent this matroid as a gammoid would be to form a complete bipartite graph with a set of vertices on one side of the bipartition, with a set of vertices on the other side of the bipartition, and with every edge directed from to . In this graph, a subset of is the set of endpoints of a set of disjoint paths if and only if it has or fewer vertices, for otherwise there aren't enough vertices in to start the paths. The special structure of this graph shows that the uniform matroid is a transversal matroid as well as being a gammoid.[5]

Alternatively, the same uniform matroid may be represented as a gammoid on a smaller graph, with only vertices, by choosing a subset of vertices and connecting each of the chosen vertices to every other vertex in the graph. Again, a subset of the vertices of the graph can be endpoints of disjoint paths if and only if it has or fewer vertices, because otherwise there are not enough vertices that can be starts of paths. In this graph, every vertex corresponds to an element of the matroid, showing that the uniform matroid is a strict gammoid.[6]

Menger's theorem and gammoid rank[edit]

The rank of a set in a gammoid defined from a graph and vertex subsets and is, by definition, the maximum number of vertex-disjoint paths from to . By Menger's theorem, it also equals the minimum cardinality of a set that intersects every path from to .[3]

Relation to transversal matroids[edit]

A transversal matroid is defined from a family of sets: its elements are the elements of the sets, and a set of these elements is independent whenever there exists a one-to-one matching of the elements of to disjoint sets containing them, called a system of distinct representatives. Equivalently, a tranversal matroid may be represented by a special kind of gammoid, defined from a directed bipartite graph that has a vertex in for each set, a vertex in for each element, and an edge from each set to each element contained in it.

Less trivially, the strict gammoids are exactly the dual matroids of the transversal matroids. To see that every strict gammoid is dual to a transversal matroid, let be a strict gammoid defined from a directed graph and starting vertex set , and consider the transversal matroid for the family of sets for each vertex , where vertex belongs to if it equals or it has an edge to . Any basis of the strict gammoid, consisting of the endpoints of some set of disjoint paths from , is the complement of a basis of the transversal matroid, matching each to the vertex such that is a path edge (or itself, if does not participate in one of the paths). Conversely every basis of the transversal matroid, consisting of a representative for each , gives rise to a complementary basis of the strict gammoid, consisting of the endpoints of the paths formed by the set of edges .[3][7]

To see, conversely, that every transversal matroid is dual to a strict gammoid, find a subfamily of the sets defining the matroid such that the subfamily has a system of distinct representatives and defines the same matroid. Form a graph that has the union of the sets as its vertices and that has an edge to the representative element of each set from the other members of the same set. Then the sets formed as above for each representative element are exactly the sets defining the original transversal matroid, so the strict gammoid formed by this graph and by the set of representative elements is dual to the given transversal matroid.[3][7]

Every gammoid is a contraction of a transversal matroid. The gammoids are the smallest class of matroids that includes the transversal matroids and is closed under duality and taking minors.[3][8][9]

Representability[edit]

It is not true that every gammoid is regular, i.e., representable over every field. In particular, the uniform matroid is not a binary matroid, and more generally the -point line can only be represented over fields with or more elements. However, every gammoid may be represented over almost every finite field.[2][3] More specifically, a gammoid with element set may be represented over every field that has at least elements.[3][10][11]

References[edit]

  1. ^ Perfect, Hazel (1968), "Applications of Menger's graph theorem", Journal of Mathematical Analysis and Applications, 22: 96–111, MR 0224494, doi:10.1016/0022-247X(68)90163-7 .
  2. ^ a b Mason, J. H. (1972), "On a class of matroids arising from paths in graphs", Proceedings of the London Mathematical Society, Third Series, 25 (1): 55–74, MR 0311496, doi:10.1112/plms/s3-25.1.55 .
  3. ^ a b c d e f g h Schrijver, Alexander (2003), Combinatorial Optimization: Polyhedra and Efficiency. Vol. B: Matroids, Trees, Stable Sets, Algorithms and Combinatorics, 24, Berlin: Springer-Verlag, pp. 659–661, ISBN 3-540-44389-4, MR 1956925 .
  4. ^ Oxley 2006, p. 100
  5. ^ Oxley, James G. (2006), Matroid Theory, Oxford Graduate Texts in Mathematics, 3, Oxford University Press, pp. 48–49, ISBN 9780199202508 .
  6. ^ Oxley (2006), p. 100.
  7. ^ a b Ingleton, A. W.; Piff, M. J. (1973), "Gammoids and transversal matroids", Journal of Combinatorial Theory, Series B, 15: 51–68, MR 0329936, doi:10.1016/0095-8956(73)90031-2 .
  8. ^ Oxley 2006, p. 115
  9. ^ Welsh, D. J. A. (2010), Matroid Theory, Courier Dover Publications, pp. 222–223, ISBN 9780486474397 .
  10. ^ Atkin, A. O. L. (1972), "Remark on a paper of Piff and Welsh", Journal of Combinatorial Theory, Series B, 13: 179–182, MR 0316281, doi:10.1016/0095-8956(72)90053-6 .
  11. ^ Lindström, Bernt (1973), "On the vector representations of induced matroids", The Bulletin of the London Mathematical Society, 5: 85–90, MR 0335313, doi:10.1112/blms/5.1.85 .