# Ends #

This file contains a definition of the ends of a simple graph, as sections of the inverse system assigning, to each finite set of vertices, the connected components of its complement.

@[reducible, inline]
abbrev SimpleGraph.ComponentCompl {V : Type u} (G : ) (K : Set V) :

The components outside a given set of vertices K

Equations
• G.ComponentCompl K = ().ConnectedComponent
Instances For
@[reducible, inline]
abbrev SimpleGraph.componentComplMk {V : Type u} {K : Set V} (G : ) {v : V} (vK : vK) :
G.ComponentCompl K

The connected component of v in G.induce Kᶜ.

Equations
• G.componentComplMk vK = ().connectedComponentMk v, vK
Instances For
def SimpleGraph.ComponentCompl.supp {V : Type u} {G : } {K : Set V} (C : G.ComponentCompl K) :
Set V

The set of vertices of G making up the connected component C

Equations
• C.supp = {v : V | ∃ (h : vK), G.componentComplMk h = C}
Instances For
theorem SimpleGraph.ComponentCompl.supp_injective {V : Type u} {G : } {K : Set V} :
Function.Injective SimpleGraph.ComponentCompl.supp
theorem SimpleGraph.ComponentCompl.supp_inj {V : Type u} {G : } {K : Set V} {C : G.ComponentCompl K} {D : G.ComponentCompl K} :
C.supp = D.supp C = D
instance SimpleGraph.ComponentCompl.setLike {V : Type u} {G : } {K : Set V} :
SetLike (G.ComponentCompl K) V
Equations
• SimpleGraph.ComponentCompl.setLike = { coe := SimpleGraph.ComponentCompl.supp, coe_injective' := }
@[simp]
theorem SimpleGraph.ComponentCompl.mem_supp_iff {V : Type u} {G : } {K : Set V} {v : V} {C : G.ComponentCompl K} :
v C ∃ (vK : vK), G.componentComplMk vK = C
theorem SimpleGraph.componentComplMk_mem {V : Type u} {K : Set V} (G : ) {v : V} (vK : vK) :
v G.componentComplMk vK
theorem SimpleGraph.componentComplMk_eq_of_adj {V : Type u} {K : Set V} (G : ) {v : V} {w : V} (vK : vK) (wK : wK) (a : G.Adj v w) :
G.componentComplMk vK = G.componentComplMk wK
instance SimpleGraph.componentCompl_nonempty_of_infinite {V : Type u} (G : ) [] (K : ) :
Nonempty (G.ComponentCompl K)

In an infinite graph, the set of components out of a finite set is nonempty.

Equations
• =
def SimpleGraph.ComponentCompl.lift {V : Type u} {G : } {K : Set V} {β : Sort u_1} (f : v : V⦄ → vKβ) (h : ∀ ⦃v w : V⦄ (hv : vK) (hw : wK), G.Adj v wf hv = f hw) :
G.ComponentCompl Kβ

A ComponentCompl specialization of Quot.lift, where soundness has to be proved only for adjacent vertices.

Equations
Instances For
theorem SimpleGraph.ComponentCompl.ind {V : Type u} {G : } {K : Set V} {β : G.ComponentCompl KProp} (f : ∀ ⦃v : V⦄ (hv : vK), β (G.componentComplMk hv)) (C : G.ComponentCompl K) :
β C
@[reducible, inline]
abbrev SimpleGraph.ComponentCompl.coeGraph {V : Type u} {G : } {K : Set V} (C : G.ComponentCompl K) :

The induced graph on the vertices C.

Equations
Instances For
theorem SimpleGraph.ComponentCompl.coe_inj {V : Type u} {G : } {K : Set V} {C : G.ComponentCompl K} {D : G.ComponentCompl K} :
C = D C = D
@[simp]
theorem SimpleGraph.ComponentCompl.nonempty {V : Type u} {G : } {K : Set V} (C : G.ComponentCompl K) :
(C).Nonempty
theorem SimpleGraph.ComponentCompl.exists_eq_mk {V : Type u} {G : } {K : Set V} (C : G.ComponentCompl K) :
∃ (v : V) (h : vK), G.componentComplMk h = C
theorem SimpleGraph.ComponentCompl.disjoint_right {V : Type u} {G : } {K : Set V} (C : G.ComponentCompl K) :
Disjoint K C
theorem SimpleGraph.ComponentCompl.not_mem_of_mem {V : Type u} {G : } {K : Set V} {C : G.ComponentCompl K} {c : V} (cC : c C) :
cK
theorem SimpleGraph.ComponentCompl.pairwise_disjoint {V : Type u} {G : } {K : Set V} :
Pairwise fun (C D : G.ComponentCompl K) => Disjoint C D
theorem SimpleGraph.ComponentCompl.mem_of_adj {V : Type u} {G : } {K : Set V} {C : G.ComponentCompl K} (c : V) (d : V) :

Any vertex adjacent to a vertex of C and not lying in K must lie in C.

theorem SimpleGraph.ComponentCompl.exists_adj_boundary_pair {V : Type u} {G : } {K : Set V} (Gc : G.Preconnected) (hK : K.Nonempty) (C : G.ComponentCompl K) :
∃ (ck : V × V), ck.1 C ck.2 K G.Adj ck.1 ck.2

Assuming G is preconnected and K not empty, given any connected component C outside of K, there exists a vertex k ∈ K adjacent to a vertex v ∈ C.

@[reducible, inline]
abbrev SimpleGraph.ComponentCompl.hom {V : Type u} {G : } {K : Set V} {L : Set V} (h : K L) (C : G.ComponentCompl L) :
G.ComponentCompl K

If K ⊆ L, the components outside of L are all contained in a single component outside of K.

Equations
Instances For
theorem SimpleGraph.ComponentCompl.subset_hom {V : Type u} {G : } {K : Set V} {L : Set V} (C : G.ComponentCompl L) (h : K L) :
C
theorem SimpleGraph.componentComplMk_mem_hom {V : Type u} {K : Set V} {L : Set V} (G : ) {v : V} (vK : vK) (h : L K) :
v SimpleGraph.ComponentCompl.hom h (G.componentComplMk vK)
theorem SimpleGraph.ComponentCompl.hom_eq_iff_le {V : Type u} {G : } {K : Set V} {L : Set V} (C : G.ComponentCompl L) (h : K L) (D : G.ComponentCompl K) :
C D
theorem SimpleGraph.ComponentCompl.hom_eq_iff_not_disjoint {V : Type u} {G : } {K : Set V} {L : Set V} (C : G.ComponentCompl L) (h : K L) (D : G.ComponentCompl K) :
¬Disjoint C D
theorem SimpleGraph.ComponentCompl.hom_refl {V : Type u} {G : } {L : Set V} (C : G.ComponentCompl L) :
theorem SimpleGraph.ComponentCompl.hom_trans {V : Type u} {G : } {K : Set V} {L : Set V} {M : Set V} (C : G.ComponentCompl L) (h : K L) (h' : M K) :
theorem SimpleGraph.ComponentCompl.hom_mk {V : Type u} {G : } {K : Set V} {L : Set V} {v : V} (vnL : vL) (h : K L) :
SimpleGraph.ComponentCompl.hom h (G.componentComplMk vnL) = G.componentComplMk
theorem SimpleGraph.ComponentCompl.hom_infinite {V : Type u} {G : } {K : Set V} {L : Set V} (C : G.ComponentCompl L) (h : K L) (Cinf : (C).Infinite) :
().Infinite
theorem SimpleGraph.ComponentCompl.infinite_iff_in_all_ranges {V : Type u} {G : } {K : } (C : G.ComponentCompl K) :
C.supp.Infinite ∀ (L : ) (h : K L), ∃ (D : G.ComponentCompl L),
instance SimpleGraph.componentCompl_finite {V : Type u} {G : } [G.LocallyFinite] [Gpc : Fact G.Preconnected] (K : ) :
Finite (G.ComponentCompl K)

For a locally finite preconnected graph, the number of components outside of any finite set is finite.

Equations
• =
@[simp]
theorem SimpleGraph.componentComplFunctor_obj {V : Type u} (G : ) (K : ()ᵒᵖ) :
G.componentComplFunctor.obj K = G.ComponentCompl K.unop
@[simp]
theorem SimpleGraph.componentComplFunctor_map {V : Type u} (G : ) :
∀ {X Y : ()ᵒᵖ} (f : X Y) (C : G.ComponentCompl X.unop), G.componentComplFunctor.map f C =

The functor assigning, to a finite set in V, the set of connected components in its complement.

Equations
• One or more equations did not get rendered due to their size.
Instances For
def SimpleGraph.end {V : Type u} (G : ) :
Set ((j : ()ᵒᵖ) → G.componentComplFunctor.obj j)

The end of a graph, defined as the sections of the functor component_compl_functor .

Equations
• G.end = G.componentComplFunctor.sections
Instances For
theorem SimpleGraph.end_hom_mk_of_mk {V : Type u} (G : ) {s : (j : ()ᵒᵖ) → G.componentComplFunctor.obj j} (sec : s G.end) {K : ()ᵒᵖ} {L : ()ᵒᵖ} (h : L K) {v : V} (vnL : vL.unop) (hs : s L = G.componentComplMk vnL) :
s K = G.componentComplMk
theorem SimpleGraph.infinite_iff_in_eventualRange {V : Type u} (G : ) {K : ()ᵒᵖ} (C : G.componentComplFunctor.obj K) :
.Infinite C G.componentComplFunctor.eventualRange K