GitHub

Labeling Elements of a Matroid

The ground set of a Matroid is always of the form {1,2,...,m} where m is a nonnegative integer. It can be useful, however, to assign descriptive labels to matroid elements. For example, when creating a matroid from a graph, the label for an element of the matroid can be a descriptor of the edge to which it corresponds.

Labels can be arbitrary and need not be unique.

The function get_label(M, x) returns the label associated with element x of the matroid M.

Default Labels

The Matroid constructors give default labels to the elements. In the case of a UniformMatroid, the label for element e is simply the integer e.

When constructing a Matroid from a matrix A, the label assigned to element e is the e-th column of A.

julia> A = [1 2 3; 4 5 6]
2×3 Matrix{Int64}:
 1  2  3
 4  5  6

julia> M = Matroid(A)
{3, 2} matroid

julia> get_label(M,1)
2-element Vector{Int64}:
 1
 4

When constructing a Matroid from a graph g, the label assigned to e is a tuple (u,v) corresponding to the e-th edge of g. 

julia> g = path_graph(5)
{5, 4} undirected simple Int64 graph

julia> M = Matroid(g)
{4, 4} matroid

julia> get_label(M,1)
(1, 2)

Setting Labels

To assign a label to element e of matroid M, use set_label(M, e, label).

The function reset_labels can be used in two ways:

  • reset_labels(M) sets the label of elemement e to be e for all e.
  • reset_labels(M, labs) sets the label of element e to be labs[e] for all e. Here, labs is a Dict.

Finding an Element with a Given Label

The function find_label(M, lab) returns an element of M whose label is lab. If no such element exists, then 0 is returned.

Note that labels may be repeated; this function returns only one element (the lowest number element) with the requested label.

Label Preservation

Matroid operations preserve labels. For example, suppose element e of matroid M has the label lab. When we construct the dual of M (which has the same ground set as M), the label assigned to e in the dual is also lab.

julia> A = [1 2 3; 4 5 6]
2×3 Matrix{Int64}:
 1  2  3
 4  5  6

julia> M = Matroid(A)
{3, 2} matroid

julia> MM = dual(M)
{3, 1} matroid

julia> get_label(M,1)
2-element Vector{Int64}:
 1
 4

julia> get_label(MM,1)
2-element Vector{Int64}:
 1
 4