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 : SimpleGraph V) (K : Set V) : Type u

The components outside a given set of vertices K

@[reducible, inline]

abbrev SimpleGraph.componentComplMk {V : Type u} {K : Set V} (G : SimpleGraph V) {v : V} (vK : v ∉ K) : G.ComponentCompl K

The connected component of v in G.induce Kᶜ.

def SimpleGraph.ComponentCompl.supp {V : Type u} {G : SimpleGraph V} {K : Set V} (C : G.ComponentCompl K) : Set V

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

theorem SimpleGraph.ComponentCompl.supp_injective {V : Type u} {G : SimpleGraph V} {K : Set V} : Function.Injective supp

theorem SimpleGraph.ComponentCompl.supp_injective_iff {V : Type u} {G : SimpleGraph V} {K : Set V} {a₁ a₂ : G.ComponentCompl K} : a₁ = a₂ ↔ a₁.supp = a₂.supp

theorem SimpleGraph.ComponentCompl.supp_inj {V : Type u} {G : SimpleGraph V} {K : Set V} {C D : G.ComponentCompl K} : C.supp = D.supp ↔ C = D

instance SimpleGraph.ComponentCompl.setLike {V : Type u} {G : SimpleGraph V} {K : Set V} : SetLike (G.ComponentCompl K) V

@[simp]

theorem SimpleGraph.ComponentCompl.mem_supp_iff {V : Type u} {G : SimpleGraph V} {K : Set V} {v : V} {C : G.ComponentCompl K} : v ∈ C ↔ ∃ (vK : v ∉ K), G.componentComplMk vK = C

theorem SimpleGraph.componentComplMk_mem {V : Type u} {K : Set V} (G : SimpleGraph V) {v : V} (vK : v ∉ K) : v ∈ G.componentComplMk vK

theorem SimpleGraph.componentComplMk_eq_of_adj {V : Type u} {K : Set V} (G : SimpleGraph V) {v w : V} (vK : v ∉ K) (wK : w ∉ K) (a : G.Adj v w) : G.componentComplMk vK = G.componentComplMk wK

instance SimpleGraph.componentCompl_nonempty_of_infinite {V : Type u} (G : SimpleGraph V) [Infinite V] (K : Finset V) : Nonempty (G.ComponentCompl ↑K)

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

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

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

theorem SimpleGraph.ComponentCompl.ind {V : Type u} {G : SimpleGraph V} {K : Set V} {β : G.ComponentCompl K → Prop} (f : ∀ ⦃v : V⦄ (hv : v ∉ K), β (G.componentComplMk hv)) (C : G.ComponentCompl K) : β C

@[reducible, inline]

abbrev SimpleGraph.ComponentCompl.coeGraph {V : Type u} {G : SimpleGraph V} {K : Set V} (C : G.ComponentCompl K) : SimpleGraph ↥C

The induced graph on the vertices C.

theorem SimpleGraph.ComponentCompl.coe_inj {V : Type u} {G : SimpleGraph V} {K : Set V} {C D : G.ComponentCompl K} : ↑C = ↑D ↔ C = D

@[simp]

theorem SimpleGraph.ComponentCompl.nonempty {V : Type u} {G : SimpleGraph V} {K : Set V} (C : G.ComponentCompl K) : (↑C).Nonempty

theorem SimpleGraph.ComponentCompl.exists_eq_mk {V : Type u} {G : SimpleGraph V} {K : Set V} (C : G.ComponentCompl K) : ∃ (v : V) (h : v ∉ K), G.componentComplMk h = C

theorem SimpleGraph.ComponentCompl.disjoint_right {V : Type u} {G : SimpleGraph V} {K : Set V} (C : G.ComponentCompl K) : Disjoint K ↑C

theorem SimpleGraph.ComponentCompl.notMem_of_mem {V : Type u} {G : SimpleGraph V} {K : Set V} {C : G.ComponentCompl K} {c : V} (cC : c ∈ C) : c ∉ K

@[deprecated SimpleGraph.ComponentCompl.notMem_of_mem (since := "2025-05-23")]

theorem SimpleGraph.ComponentCompl.not_mem_of_mem {V : Type u} {G : SimpleGraph V} {K : Set V} {C : G.ComponentCompl K} {c : V} (cC : c ∈ C) : c ∉ K

Alias of SimpleGraph.ComponentCompl.notMem_of_mem.

theorem SimpleGraph.ComponentCompl.pairwise_disjoint {V : Type u} {G : SimpleGraph V} {K : Set V} : Pairwise fun (C D : G.ComponentCompl K) => Disjoint ↑C ↑D

theorem SimpleGraph.ComponentCompl.mem_of_adj {V : Type u} {G : SimpleGraph V} {K : Set V} {C : G.ComponentCompl K} (c d : V) : c ∈ C → d ∉ K → G.Adj c d → d ∈ C

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 : SimpleGraph V} {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 : SimpleGraph V} {K 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.

theorem SimpleGraph.ComponentCompl.subset_hom {V : Type u} {G : SimpleGraph V} {K L : Set V} (C : G.ComponentCompl L) (h : K ⊆ L) : ↑C ⊆ ↑(hom h C)

theorem SimpleGraph.componentComplMk_mem_hom {V : Type u} {K L : Set V} (G : SimpleGraph V) {v : V} (vK : v ∉ K) (h : L ⊆ K) : v ∈ ComponentCompl.hom h (G.componentComplMk vK)

theorem SimpleGraph.ComponentCompl.hom_eq_iff_le {V : Type u} {G : SimpleGraph V} {K L : Set V} (C : G.ComponentCompl L) (h : K ⊆ L) (D : G.ComponentCompl K) : hom h C = D ↔ ↑C ⊆ ↑D

theorem SimpleGraph.ComponentCompl.hom_eq_iff_not_disjoint {V : Type u} {G : SimpleGraph V} {K L : Set V} (C : G.ComponentCompl L) (h : K ⊆ L) (D : G.ComponentCompl K) : hom h C = D ↔ ¬Disjoint ↑C ↑D

theorem SimpleGraph.ComponentCompl.hom_refl {V : Type u} {G : SimpleGraph V} {L : Set V} (C : G.ComponentCompl L) : hom ⋯ C = C

theorem SimpleGraph.ComponentCompl.hom_trans {V : Type u} {G : SimpleGraph V} {K L M : Set V} (C : G.ComponentCompl L) (h : K ⊆ L) (h' : M ⊆ K) : hom ⋯ C = hom h' (hom h C)

theorem SimpleGraph.ComponentCompl.hom_mk {V : Type u} {G : SimpleGraph V} {K L : Set V} {v : V} (vnL : v ∉ L) (h : K ⊆ L) : hom h (G.componentComplMk vnL) = G.componentComplMk ⋯

theorem SimpleGraph.ComponentCompl.hom_infinite {V : Type u} {G : SimpleGraph V} {K L : Set V} (C : G.ComponentCompl L) (h : K ⊆ L) (Cinf : (↑C).Infinite) : (↑(hom h C)).Infinite

theorem SimpleGraph.ComponentCompl.infinite_iff_in_all_ranges {V : Type u} {G : SimpleGraph V} {K : Finset V} (C : G.ComponentCompl ↑K) : C.supp.Infinite ↔ ∀ (L : Finset V) (h : K ⊆ L), ∃ (D : G.ComponentCompl ↑L), hom h D = C

instance SimpleGraph.componentCompl_finite {V : Type u} {G : SimpleGraph V} [G.LocallyFinite] [Gpc : Fact G.Preconnected] (K : Finset V) : Finite (G.ComponentCompl ↑K)

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

def SimpleGraph.componentComplFunctor {V : Type u} (G : SimpleGraph V) : CategoryTheory.Functor (Finset V)ᵒᵖ (Type u)

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

@[simp]

theorem SimpleGraph.componentComplFunctor_obj {V : Type u} (G : SimpleGraph V) (K : (Finset V)ᵒᵖ) : G.componentComplFunctor.obj K = G.ComponentCompl ↑(Opposite.unop K)

@[simp]

theorem SimpleGraph.componentComplFunctor_map {V : Type u} (G : SimpleGraph V) {X✝ Y✝ : (Finset V)ᵒᵖ} (f : X✝ ⟶ Y✝) (C : G.ComponentCompl ↑(Opposite.unop X✝)) : G.componentComplFunctor.map f C = ComponentCompl.hom ⋯ C

def SimpleGraph.end {V : Type u} (G : SimpleGraph V) : Set ((j : (Finset V)ᵒᵖ) → G.componentComplFunctor.obj j)

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

theorem SimpleGraph.end_hom_mk_of_mk {V : Type u} (G : SimpleGraph V) {s : (j : (Finset V)ᵒᵖ) → G.componentComplFunctor.obj j} (sec : s ∈ G.end) {K L : (Finset V)ᵒᵖ} (h : L ⟶ K) {v : V} (vnL : v ∉ Opposite.unop L) (hs : s L = G.componentComplMk vnL) : s K = G.componentComplMk ⋯

theorem SimpleGraph.infinite_iff_in_eventualRange {V : Type u} (G : SimpleGraph V) {K : (Finset V)ᵒᵖ} (C : G.componentComplFunctor.obj K) : (ComponentCompl.supp C).Infinite ↔ C ∈ G.componentComplFunctor.eventualRange K