Changes
Let ''G=(V,E)'' be an undirected graph with no loops. Let ''l'' be an integer, and <math>m: V \to {\mathbb Z}_+</math> a function on the nodes for which <math>m(u)+m(v) \geq l</math> for every <math>uv \in E</math>. The '''count matroid''' ''M=M(G,l,m)'' is defined on ground set ''E'': an edge-set <math>F \subseteq E</math> is independent if and only if
Count matroids were introduced (with a more restricted definition) by White <ref>N. L. White, ''A pruning theorem for linear count matroids'', Congr. Numer. 54 (1986), 259-264.</ref> and generalized by Whiteley <ref>W. Whiteley, ''Some matroids from discrete applied geometry'', in Contemporary Mathematics, J. G. Oxley, J. E. Bonin and B. Servatius, Eds., vol. 197. American Mathematical Society, 1996.</ref>, who also extended the definition to hypergraphs.
Let ''H=(V,E)'' be a hypergraph, ''l'' an integer, and <math>m: V \to {\mathbb Z}_+</math> a function on the nodes for which <math>m(e) \geq l</math> for every <math>e \in E</math>. The '''ground set of the count matroid''' ''M=M(H,l,m)'' is defined on ground set ''E'': , and a hyperedge-set <math>F \subseteq E</math> is independent if and only if
<math>i_F(X)\leq m(X)-l \text{ for every } X \subseteq V \text{ with } i_F(X) \geq 1.</math>
* If ''l''=1 and <math>m \equiv 1</math> then we get the cycle matroid of ''G''.
* If ''l=k'' and <math>m \equiv k</math> for some positive integer ''k'' then by [[Nash-Williams' forest cover theorem]] the independent sets are the edge-sets that can be partitioned into ''k'' forests.
* If ''l=3'' and <math>m \equiv 2</math> then we get the [[Rigidity matrix|2-dimensional rigidity matroid]] of the graph ''G'' by [[Laman's theorem]], see also the page [[Rigid graph]].* A [[transversal matroid]] defined by a bipartite graph ''G=(S,T;E)'' corresponds to ''M(H,l,m)'' with the parameters <math>m \equiv 1</math> and ''l=0'', where ''H'' is the hypergraph representation of ''G'' with node set ''T''.* The [[hypergraphic matroid]] of a hypergraph ''H''is ''M(H,l,m)'' with the parameters <math>m \equiv 1</math> and ''l=1''.
==Directed graphs==
