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Mathlib.Combinatorics.Quiver.ConnectedComponent

Weakly connected components #

For a quiver V, define the type WeaklyConnectedComponent V as the quotient of V by the relation which identifies a with b if there is a path from a to b in Symmetrify V. (These zigzags can be seen as a proof-relevant analogue of EqvGen.)

Strongly connected components have not yet been defined.

def Quiver.zigzagSetoid (V : Type u_1) [inst : Quiver V] :

Two vertices are related in the zigzag setoid if there is a zigzag of arrows from one to the other.

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def Quiver.WeaklyConnectedComponent (V : Type u_1) [inst : Quiver V] :
Type u_1

The type of weakly connected components of a directed graph. Two vertices are in the same weakly connected component if there is a zigzag of arrows from one to the other.

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The weakly connected component corresponding to a vertex.

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  • Quiver.WeaklyConnectedComponent.mk = Quotient.mk'
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  • Quiver.WeaklyConnectedComponent.instCoeTCWeaklyConnectedComponent = { coe := Quiver.WeaklyConnectedComponent.mk }
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A wide subquiver H of Symmetrify V determines a wide subquiver of V, containing an an arrow e if either e or its reversal is in H.

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