def Multigraph.graphicMatroid {V : Type u_1} {E : Type u_2} [Finite E] (G : Multigraph V E) : Matroid E

The cycle (graphic) matroid of a finite multigraph G: the matroid on the edge set E whose independent sets are the forests of G.

Equations

• G.graphicMatroid = (Multigraph.indepMatroid✝ G).matroid

@[simp]

theorem Multigraph.graphicMatroid_indep {V : Type u_1} {E : Type u_2} [Finite E] (G : Multigraph V E) {I : Set E} : G.graphicMatroid.Indep I ↔ G.IsForest I

@[simp]

theorem Multigraph.graphicMatroid_E {V : Type u_1} {E : Type u_2} [Finite E] (G : Multigraph V E) : G.graphicMatroid.E = Set.univ