mathlib documentation

combinatorics.simple_graph.connectivity

Graph connectivity #

In a simple graph,

Warning: graph theorists mean something different by "path" than do homotopy theorists. A "walk" in graph theory is a "path" in homotopy theory. Another warning: some graph theorists use "path" and "simple path" for "walk" and "path."

Some definitions and theorems have inspiration from multigraph counterparts in [Cho94].

Main definitions #

Tags #

walks, trails, paths, circuits, cycles

source
inductive simple_graph.walk {V : Type u} (G : simple_graph V) :
V → V → Type u

A walk is a sequence of adjacent vertices. For vertices u v : V, the type walk u v consists of all walks starting at u and ending at v.

We say that a walk visits the vertices it contains. The set of vertices a walk visits is simple_graph.walk.support.

See simple_graph.walk.nil' and simple_graph.walk.cons' for patterns that can be useful in definitions since they make the vertices explicit.

Instances for simple_graph.walk
source
@[protected, instance]
def simple_graph.walk.decidable_eq {V : Type u} [a : decidable_eq V] (G : simple_graph V) (ᾰ ᾰ_1 : V) :
decidable_eq (G.walk ᾰ ᾰ_1)
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@[protected, instance]
def simple_graph.walk.inhabited {V : Type u} (G : simple_graph V) (v : V) :
inhabited (G.walk v v)
Equations
source
@[simp]
theorem simple_graph.walk.inhabited_default {V : Type u} (G : simple_graph V) (v : V) :
inhabited.default = simple_graph.walk.nil
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@[reducible]
def simple_graph.walk.nil' {V : Type u} {G : simple_graph V} (u : V) :
G.walk u u

Pattern to get walk.nil with the vertex as an explicit argument.

source
@[reducible]
def simple_graph.walk.cons' {V : Type u} {G : simple_graph V} (u v w : V) (h : G.adj u v) (p : G.walk v w) :
G.walk u w

Pattern to get walk.cons with the vertices as explicit arguments.

source
theorem simple_graph.walk.exists_eq_cons_of_ne {V : Type u} {G : simple_graph V} {u v : V} (hne : u ≠ v) (p : G.walk u v) :
∃ (w : V) (h : G.adj u w) (p' : G.walk w v), p = simple_graph.walk.cons h p'
source
def simple_graph.walk.length {V : Type u} {G : simple_graph V} {u v : V} :
G.walk u v → ℕ

The length of a walk is the number of edges/darts along it.

Equations
source
@[trans]
def simple_graph.walk.append {V : Type u} {G : simple_graph V} {u v w : V} :
G.walk u v → G.walk v w → G.walk u w

The concatenation of two compatible walks.

Equations
source
@[protected]
def simple_graph.walk.reverse_aux {V : Type u} {G : simple_graph V} {u v w : V} :
G.walk u v → G.walk u w → G.walk v w

The concatenation of the reverse of the first walk with the second walk.

Equations
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@[symm]
def simple_graph.walk.reverse {V : Type u} {G : simple_graph V} {u v : V} (w : G.walk u v) :
G.walk v u

The walk in reverse.

Equations
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def simple_graph.walk.get_vert {V : Type u} {G : simple_graph V} {u v : V} (p : G.walk u v) (n : ℕ) :
V

Get the nth vertex from a walk, where n is generally expected to be between 0 and p.length, inclusive. If n is greater than or equal to p.length, the result is the path's endpoint.

Equations
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@[simp]
theorem simple_graph.walk.get_vert_zero {V : Type u} {G : simple_graph V} {u v : V} (w : G.walk u v) :
w.get_vert 0 = u
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theorem simple_graph.walk.get_vert_of_length_le {V : Type u} {G : simple_graph V} {u v : V} (w : G.walk u v) {i : ℕ} (hi : w.length ≤ i) :
w.get_vert i = v
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@[simp]
theorem simple_graph.walk.get_vert_length {V : Type u} {G : simple_graph V} {u v : V} (w : G.walk u v) :
w.get_vert w.length = v
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theorem simple_graph.walk.adj_get_vert_succ {V : Type u} {G : simple_graph V} {u v : V} (w : G.walk u v) {i : ℕ} (hi : i < w.length) :
G.adj (w.get_vert i) (w.get_vert (i + 1))
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@[simp]
theorem simple_graph.walk.cons_append {V : Type u} {G : simple_graph V} {u v w x : V} (h : G.adj u v) (p : G.walk v w) (q : G.walk w x) :
(simple_graph.walk.cons h p).append q = simple_graph.walk.cons h (p.append q)
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@[simp]
theorem simple_graph.walk.cons_nil_append {V : Type u} {G : simple_graph V} {u v w : V} (h : G.adj u v) (p : G.walk v w) :
(simple_graph.walk.cons h simple_graph.walk.nil).append p = simple_graph.walk.cons h p
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@[simp]
theorem simple_graph.walk.append_nil {V : Type u} {G : simple_graph V} {u v : V} (p : G.walk u v) :
p.append simple_graph.walk.nil = p
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@[simp]
theorem simple_graph.walk.nil_append {V : Type u} {G : simple_graph V} {u v : V} (p : G.walk u v) :
simple_graph.walk.nil.append p = p
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theorem simple_graph.walk.append_assoc {V : Type u} {G : simple_graph V} {u v w x : V} (p : G.walk u v) (q : G.walk v w) (r : G.walk w x) :
p.append (q.append r) = (p.append q).append r
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@[simp]
theorem simple_graph.walk.reverse_nil {V : Type u} {G : simple_graph V} {u : V} :
simple_graph.walk.nil.reverse = simple_graph.walk.nil
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theorem simple_graph.walk.reverse_singleton {V : Type u} {G : simple_graph V} {u v : V} (h : G.adj u v) :
(simple_graph.walk.cons h simple_graph.walk.nil).reverse = simple_graph.walk.cons _ simple_graph.walk.nil
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@[simp]
theorem simple_graph.walk.cons_reverse_aux {V : Type u} {G : simple_graph V} {u v w x : V} (p : G.walk u v) (q : G.walk w x) (h : G.adj w u) :
(simple_graph.walk.cons h p).reverse_aux q = p.reverse_aux (simple_graph.walk.cons _ q)
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@[protected, simp]
theorem simple_graph.walk.append_reverse_aux {V : Type u} {G : simple_graph V} {u v w x : V} (p : G.walk u v) (q : G.walk v w) (r : G.walk u x) :
(p.append q).reverse_aux r = q.reverse_aux (p.reverse_aux r)
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@[protected, simp]
theorem simple_graph.walk.reverse_aux_append {V : Type u} {G : simple_graph V} {u v w x : V} (p : G.walk u v) (q : G.walk u w) (r : G.walk w x) :
(p.reverse_aux q).append r = p.reverse_aux (q.append r)
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@[protected]
theorem simple_graph.walk.reverse_aux_eq_reverse_append {V : Type u} {G : simple_graph V} {u v w : V} (p : G.walk u v) (q : G.walk u w) :
p.reverse_aux q = p.reverse.append q
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@[simp]
theorem simple_graph.walk.reverse_cons {V : Type u} {G : simple_graph V} {u v w : V} (h : G.adj u v) (p : G.walk v w) :
(simple_graph.walk.cons h p).reverse = p.reverse.append (simple_graph.walk.cons _ simple_graph.walk.nil)
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@[simp]
theorem simple_graph.walk.reverse_append {V : Type u} {G : simple_graph V} {u v w : V} (p : G.walk u v) (q : G.walk v w) :
(p.append q).reverse = q.reverse.append p.reverse
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@[simp]
theorem simple_graph.walk.reverse_reverse {V : Type u} {G : simple_graph V} {u v : V} (p : G.walk u v) :
p.reverse.reverse = p
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@[simp]
theorem simple_graph.walk.length_nil {V : Type u} {G : simple_graph V} {u : V} :
simple_graph.walk.nil.length = 0
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@[simp]
theorem simple_graph.walk.length_cons {V : Type u} {G : simple_graph V} {u v w : V} (h : G.adj u v) (p : G.walk v w) :
(simple_graph.walk.cons h p).length = p.length + 1
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@[simp]
theorem simple_graph.walk.length_append {V : Type u} {G : simple_graph V} {u v w : V} (p : G.walk u v) (q : G.walk v w) :
(p.append q).length = p.length + q.length
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@[protected, simp]
theorem simple_graph.walk.length_reverse_aux {V : Type u} {G : simple_graph V} {u v w : V} (p : G.walk u v) (q : G.walk u w) :
(p.reverse_aux q).length = p.length + q.length
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@[simp]
theorem simple_graph.walk.length_reverse {V : Type u} {G : simple_graph V} {u v : V} (p : G.walk u v) :
p.reverse.length = p.length
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theorem simple_graph.walk.eq_of_length_eq_zero {V : Type u} {G : simple_graph V} {u v : V} {p : G.walk u v} :
p.length = 0 → u = v
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@[simp]
theorem simple_graph.walk.exists_length_eq_zero_iff {V : Type u} {G : simple_graph V} {u v : V} :
(∃ (p : G.walk u v), p.length = 0) ↔ u = v
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@[simp]
theorem simple_graph.walk.length_eq_zero_iff {V : Type u} {G : simple_graph V} {u : V} {p : G.walk u u} :
p.length = 0 ↔ p = simple_graph.walk.nil
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def simple_graph.walk.support {V : Type u} {G : simple_graph V} {u v : V} :
G.walk u v → list V

The support of a walk is the list of vertices it visits in order.

Equations
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def simple_graph.walk.darts {V : Type u} {G : simple_graph V} {u v : V} :
G.walk u v → list G.dart

The darts of a walk is the list of darts it visits in order.

Equations
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def simple_graph.walk.edges {V : Type u} {G : simple_graph V} {u v : V} (p : G.walk u v) :
list (sym2 V)

The edges of a walk is the list of edges it visits in order. This is defined to be the list of edges underlying simple_graph.walk.darts.

Equations
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@[simp]
theorem simple_graph.walk.support_nil {V : Type u} {G : simple_graph V} {u : V} :
simple_graph.walk.nil.support = [u]
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@[simp]
theorem simple_graph.walk.support_cons {V : Type u} {G : simple_graph V} {u v w : V} (h : G.adj u v) (p : G.walk v w) :
(simple_graph.walk.cons h p).support = u :: p.support
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theorem simple_graph.walk.support_append {V : Type u} {G : simple_graph V} {u v w : V} (p : G.walk u v) (p' : G.walk v w) :
(p.append p').support = p.support ++ p'.support.tail
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@[simp]
theorem simple_graph.walk.support_reverse {V : Type u} {G : simple_graph V} {u v : V} (p : G.walk u v) :
p.reverse.support = p.support.reverse
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theorem simple_graph.walk.support_ne_nil {V : Type u} {G : simple_graph V} {u v : V} (p : G.walk u v) :
p.support ≠ list.nil
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theorem simple_graph.walk.tail_support_append {V : Type u} {G : simple_graph V} {u v w : V} (p : G.walk u v) (p' : G.walk v w) :
(p.append p').support.tail = p.support.tail ++ p'.support.tail
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theorem simple_graph.walk.support_eq_cons {V : Type u} {G : simple_graph V} {u v : V} (p : G.walk u v) :
p.support = u :: p.support.tail
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@[simp]
theorem simple_graph.walk.start_mem_support {V : Type u} {G : simple_graph V} {u v : V} (p : G.walk u v) :
u ∈ p.support
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@[simp]
theorem simple_graph.walk.end_mem_support {V : Type u} {G : simple_graph V} {u v : V} (p : G.walk u v) :
v ∈ p.support
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theorem simple_graph.walk.mem_support_iff {V : Type u} {G : simple_graph V} {u v w : V} (p : G.walk u v) :
w ∈ p.support ↔ w = u ∨ w ∈ p.support.tail
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theorem simple_graph.walk.mem_support_nil_iff {V : Type u} {G : simple_graph V} {u v : V} :
u ∈ simple_graph.walk.nil.support ↔ u = v
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@[simp]
theorem simple_graph.walk.mem_tail_support_append_iff {V : Type u} {G : simple_graph V} {t u v w : V} (p : G.walk u v) (p' : G.walk v w) :
t ∈ (p.append p').support.tail ↔ t ∈ p.support.tail ∨ t ∈ p'.support.tail
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@[simp]
theorem simple_graph.walk.end_mem_tail_support_of_ne {V : Type u} {G : simple_graph V} {u v : V} (h : u ≠ v) (p : G.walk u v) :
v ∈ p.support.tail
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@[simp]
theorem simple_graph.walk.mem_support_append_iff {V : Type u} {G : simple_graph V} {t u v w : V} (p : G.walk u v) (p' : G.walk v w) :
t ∈ (p.append p').support ↔ t ∈ p.support ∨ t ∈ p'.support
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@[simp]
theorem simple_graph.walk.subset_support_append_left {V : Type u} {G : simple_graph V} {u v w : V} (p : G.walk u v) (q : G.walk v w) :
p.support ⊆ (p.append q).support
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@[simp]
theorem simple_graph.walk.subset_support_append_right {V : Type u} {G : simple_graph V} {u v w : V} (p : G.walk u v) (q : G.walk v w) :
q.support ⊆ (p.append q).support
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theorem simple_graph.walk.coe_support {V : Type u} {G : simple_graph V} {u v : V} (p : G.walk u v) :
↑(p.support) = {u} + ↑(p.support.tail)
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theorem simple_graph.walk.coe_support_append {V : Type u} {G : simple_graph V} {u v w : V} (p : G.walk u v) (p' : G.walk v w) :
↑((p.append p').support) = {u} + ↑(p.support.tail) + ↑(p'.support.tail)
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theorem simple_graph.walk.coe_support_append' {V : Type u} {G : simple_graph V} [decidable_eq V] {u v w : V} (p : G.walk u v) (p' : G.walk v w) :
↑((p.append p').support) = ↑(p.support) + ↑(p'.support) - {v}
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theorem simple_graph.walk.chain_adj_support {V : Type u} {G : simple_graph V} {u v w : V} (h : G.adj u v) (p : G.walk v w) :
list.chain G.adj u p.support
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theorem simple_graph.walk.chain'_adj_support {V : Type u} {G : simple_graph V} {u v : V} (p : G.walk u v) :
list.chain' G.adj p.support
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theorem simple_graph.walk.chain_dart_adj_darts {V : Type u} {G : simple_graph V} {d : G.dart} {v w : V} (h : d.to_prod.snd = v) (p : G.walk v w) :
list.chain G.dart_adj d p.darts
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theorem simple_graph.walk.chain'_dart_adj_darts {V : Type u} {G : simple_graph V} {u v : V} (p : G.walk u v) :
list.chain' G.dart_adj p.darts
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theorem simple_graph.walk.edges_subset_edge_set {V : Type u} {G : simple_graph V} {u v : V} (p : G.walk u v) ⦃e : sym2 V⦄ (h : e ∈ p.edges) :
e ∈ G.edge_set

Every edge in a walk's edge list is an edge of the graph. It is written in this form (rather than using ⊆) to avoid unsightly coercions.

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@[simp]
theorem simple_graph.walk.darts_nil {V : Type u} {G : simple_graph V} {u : V} :
simple_graph.walk.nil.darts = list.nil
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@[simp]
theorem simple_graph.walk.darts_cons {V : Type u} {G : simple_graph V} {u v w : V} (h : G.adj u v) (p : G.walk v w) :
(simple_graph.walk.cons h p).darts = {to_prod := (u, v), is_adj := h} :: p.darts
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@[simp]
theorem simple_graph.walk.darts_append {V : Type u} {G : simple_graph V} {u v w : V} (p : G.walk u v) (p' : G.walk v w) :
(p.append p').darts = p.darts ++ p'.darts
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@[simp]
theorem simple_graph.walk.darts_reverse {V : Type u} {G : simple_graph V} {u v : V} (p : G.walk u v) :
p.reverse.darts = (list.map simple_graph.dart.symm p.darts).reverse
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theorem simple_graph.walk.mem_darts_reverse {V : Type u} {G : simple_graph V} {u v : V} {d : G.dart} {p : G.walk u v} :
d ∈ p.reverse.darts ↔ d.symm ∈ p.darts
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theorem simple_graph.walk.cons_map_snd_darts {V : Type u} {G : simple_graph V} {u v : V} (p : G.walk u v) :
u :: list.map simple_graph.dart.snd p.darts = p.support
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theorem simple_graph.walk.map_snd_darts {V : Type u} {G : simple_graph V} {u v : V} (p : G.walk u v) :
list.map simple_graph.dart.snd p.darts = p.support.tail
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theorem simple_graph.walk.map_fst_darts_append {V : Type u} {G : simple_graph V} {u v : V} (p : G.walk u v) :
list.map simple_graph.dart.fst p.darts ++ [v] = p.support
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theorem simple_graph.walk.map_fst_darts {V : Type u} {G : simple_graph V} {u v : V} (p : G.walk u v) :
list.map simple_graph.dart.fst p.darts = p.support.init
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@[simp]
theorem simple_graph.walk.edges_nil {V : Type u} {G : simple_graph V} {u : V} :
simple_graph.walk.nil.edges = list.nil
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@[simp]
theorem simple_graph.walk.edges_cons {V : Type u} {G : simple_graph V} {u v w : V} (h : G.adj u v) (p : G.walk v w) :
(simple_graph.walk.cons h p).edges = ⟦(u, v)⟧ :: p.edges
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@[simp]
theorem simple_graph.walk.edges_append {V : Type u} {G : simple_graph V} {u v w : V} (p : G.walk u v) (p' : G.walk v w) :
(p.append p').edges = p.edges ++ p'.edges
source
@[simp]
theorem simple_graph.walk.edges_reverse {V : Type u} {G : simple_graph V} {u v : V} (p : G.walk u v) :
p.reverse.edges = p.edges.reverse
source
@[simp]
theorem simple_graph.walk.length_support {V : Type u} {G : simple_graph V} {u v : V} (p : G.walk u v) :
p.support.length = p.length + 1
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@[simp]
theorem simple_graph.walk.length_darts {V : Type u} {G : simple_graph V} {u v : V} (p : G.walk u v) :
p.darts.length = p.length
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@[simp]
theorem simple_graph.walk.length_edges {V : Type u} {G : simple_graph V} {u v : V} (p : G.walk u v) :
p.edges.length = p.length
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theorem simple_graph.walk.dart_fst_mem_support_of_mem_darts {V : Type u} {G : simple_graph V} {u v : V} (p : G.walk u v) {d : G.dart} :
d ∈ p.darts → d.to_prod.fst ∈ p.support
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theorem simple_graph.walk.dart_snd_mem_support_of_mem_darts {V : Type u} {G : simple_graph V} {u v : V} (p : G.walk u v) {d : G.dart} (h : d ∈ p.darts) :
d.to_prod.snd ∈ p.support
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theorem simple_graph.walk.fst_mem_support_of_mem_edges {V : Type u} {G : simple_graph V} {t u v w : V} (p : G.walk v w) (he : ⟦(t, u)⟧ ∈ p.edges) :
t ∈ p.support
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theorem simple_graph.walk.snd_mem_support_of_mem_edges {V : Type u} {G : simple_graph V} {t u v w : V} (p : G.walk v w) (he : ⟦(t, u)⟧ ∈ p.edges) :
u ∈ p.support
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theorem simple_graph.walk.darts_nodup_of_support_nodup {V : Type u} {G : simple_graph V} {u v : V} {p : G.walk u v} (h : p.support.nodup) :
p.darts.nodup
source
theorem simple_graph.walk.edges_nodup_of_support_nodup {V : Type u} {G : simple_graph V} {u v : V} {p : G.walk u v} (h : p.support.nodup) :
p.edges.nodup

Trails, paths, circuits, cycles #

source
structure simple_graph.walk.is_trail {V : Type u} {G : simple_graph V} {u v : V} (p : G.walk u v) :
Prop

A trail is a walk with no repeating edges.

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structure simple_graph.walk.is_path {V : Type u} {G : simple_graph V} {u v : V} (p : G.walk u v) :
Prop

A path is a walk with no repeating vertices. Use simple_graph.walk.is_path.mk' for a simpler constructor.

source
structure simple_graph.walk.is_circuit {V : Type u} {G : simple_graph V} {u : V} (p : G.walk u u) :
Prop

A circuit at u : V is a nonempty trail beginning and ending at u.

source
structure simple_graph.walk.is_cycle {V : Type u} {G : simple_graph V} {u : V} (p : G.walk u u) :
Prop

A cycle at u : V is a circuit at u whose only repeating vertex is u (which appears exactly twice).

source
theorem simple_graph.walk.is_trail_def {V : Type u} {G : simple_graph V} {u v : V} (p : G.walk u v) :
p.is_trail ↔ p.edges.nodup
source
theorem simple_graph.walk.is_path.mk' {V : Type u} {G : simple_graph V} {u v : V} {p : G.walk u v} (h : p.support.nodup) :
p.is_path
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theorem simple_graph.walk.is_path_def {V : Type u} {G : simple_graph V} {u v : V} (p : G.walk u v) :
p.is_path ↔ p.support.nodup
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theorem simple_graph.walk.is_cycle_def {V : Type u} {G : simple_graph V} {u : V} (p : G.walk u u) :
p.is_cycle ↔ p.is_trail ∧ p ≠ simple_graph.walk.nil ∧ p.support.tail.nodup
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@[simp]
theorem simple_graph.walk.is_trail.nil {V : Type u} {G : simple_graph V} {u : V} :
simple_graph.walk.nil.is_trail
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theorem simple_graph.walk.is_trail.of_cons {V : Type u} {G : simple_graph V} {u v w : V} {h : G.adj u v} {p : G.walk v w} :
(simple_graph.walk.cons h p).is_trail → p.is_trail
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@[simp]
theorem simple_graph.walk.cons_is_trail_iff {V : Type u} {G : simple_graph V} {u v w : V} (h : G.adj u v) (p : G.walk v w) :
(simple_graph.walk.cons h p).is_trail ↔ p.is_trail ∧ ⟦(u, v)⟧ ∉ p.edges
source
theorem simple_graph.walk.is_trail.reverse {V : Type u} {G : simple_graph V} {u v : V} (p : G.walk u v) (h : p.is_trail) :
p.reverse.is_trail
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@[simp]
theorem simple_graph.walk.reverse_is_trail_iff {V : Type u} {G : simple_graph V} {u v : V} (p : G.walk u v) :
p.reverse.is_trail ↔ p.is_trail
source
theorem simple_graph.walk.is_trail.of_append_left {V : Type u} {G : simple_graph V} {u v w : V} {p : G.walk u v} {q : G.walk v w} (h : (p.append q).is_trail) :
p.is_trail
source
theorem simple_graph.walk.is_trail.of_append_right {V : Type u} {G : simple_graph V} {u v w : V} {p : G.walk u v} {q : G.walk v w} (h : (p.append q).is_trail) :
q.is_trail
source
theorem simple_graph.walk.is_trail.count_edges_le_one {V : Type u} {G : simple_graph V} [decidable_eq V] {u v : V} {p : G.walk u v} (h : p.is_trail) (e : sym2 V) :
list.count e p.edges ≤ 1
source
theorem simple_graph.walk.is_trail.count_edges_eq_one {V : Type u} {G : simple_graph V} [decidable_eq V] {u v : V} {p : G.walk u v} (h : p.is_trail) {e : sym2 V} (he : e ∈ p.edges) :
list.count e p.edges = 1
source
@[simp]
theorem simple_graph.walk.is_path.nil {V : Type u} {G : simple_graph V} {u : V} :
simple_graph.walk.nil.is_path
source
theorem simple_graph.walk.is_path.of_cons {V : Type u} {G : simple_graph V} {u v w : V} {h : G.adj u v} {p : G.walk v w} :
(simple_graph.walk.cons h p).is_path → p.is_path
source
@[simp]
theorem simple_graph.walk.cons_is_path_iff {V : Type u} {G : simple_graph V} {u v w : V} (h : G.adj u v) (p : G.walk v w) :
(simple_graph.walk.cons h p).is_path ↔ p.is_path ∧ u ∉ p.support
source
theorem simple_graph.walk.is_path.reverse {V : Type u} {G : simple_graph V} {u v : V} {p : G.walk u v} (h : p.is_path) :
p.reverse.is_path
source
@[simp]
theorem simple_graph.walk.is_path_reverse_iff {V : Type u} {G : simple_graph V} {u v : V} (p : G.walk u v) :
p.reverse.is_path ↔ p.is_path
source
theorem simple_graph.walk.is_path.of_append_left {V : Type u} {G : simple_graph V} {u v w : V} {p : G.walk u v} {q : G.walk v w} :
(p.append q).is_path → p.is_path
source
theorem simple_graph.walk.is_path.of_append_right {V : Type u} {G : simple_graph V} {u v w : V} {p : G.walk u v} {q : G.walk v w} (h : (p.append q).is_path) :
q.is_path
source
@[simp]
theorem simple_graph.walk.is_cycle.not_of_nil {V : Type u} {G : simple_graph V} {u : V} :
¬simple_graph.walk.nil.is_cycle

Walk decompositions #

source
def simple_graph.walk.take_until {V : Type u} {G : simple_graph V} [decidable_eq V] {v w : V} (p : G.walk v w) (u : V) (h : u ∈ p.support) :
G.walk v u

Given a vertex in the support of a path, give the path up until (and including) that vertex.

Equations
source
def simple_graph.walk.drop_until {V : Type u} {G : simple_graph V} [decidable_eq V] {v w : V} (p : G.walk v w) (u : V) (h : u ∈ p.support) :
G.walk u w

Given a vertex in the support of a path, give the path from (and including) that vertex to the end. In other words, drop vertices from the front of a path until (and not including) that vertex.

Equations
source
@[simp]
theorem simple_graph.walk.take_spec {V : Type u} {G : simple_graph V} [decidable_eq V] {u v w : V} (p : G.walk v w) (h : u ∈ p.support) :
(p.take_until u h).append (p.drop_until u h) = p

The take_until and drop_until functions split a walk into two pieces. The lemma count_support_take_until_eq_one specifies where this split occurs.

source
theorem simple_graph.walk.mem_support_iff_exists_append {V : Type u} {G : simple_graph V} {u v w : V} {p : G.walk u v} :
w ∈ p.support ↔ ∃ (q : G.walk u w) (r : G.walk w v), p = q.append r
source
@[simp]
theorem simple_graph.walk.count_support_take_until_eq_one {V : Type u} {G : simple_graph V} [decidable_eq V] {u v w : V} (p : G.walk v w) (h : u ∈ p.support) :
list.count u (p.take_until u h).support = 1
source
theorem simple_graph.walk.count_edges_take_until_le_one {V : Type u} {G : simple_graph V} [decidable_eq V] {u v w : V} (p : G.walk v w) (h : u ∈ p.support) (x : V) :
list.count ⟦(u, x)⟧ (p.take_until u h).edges ≤ 1
source
theorem simple_graph.walk.support_take_until_subset {V : Type u} {G : simple_graph V} [decidable_eq V] {u v w : V} (p : G.walk v w) (h : u ∈ p.support) :
(p.take_until u h).support ⊆ p.support
source
theorem simple_graph.walk.support_drop_until_subset {V : Type u} {G : simple_graph V} [decidable_eq V] {u v w : V} (p : G.walk v w) (h : u ∈ p.support) :
(p.drop_until u h).support ⊆ p.support
source
theorem simple_graph.walk.darts_take_until_subset {V : Type u} {G : simple_graph V} [decidable_eq V] {u v w : V} (p : G.walk v w) (h : u ∈ p.support) :
(p.take_until u h).darts ⊆ p.darts
source
theorem simple_graph.walk.darts_drop_until_subset {V : Type u} {G : simple_graph V} [decidable_eq V] {u v w : V} (p : G.walk v w) (h : u ∈ p.support) :
(p.drop_until u h).darts ⊆ p.darts
source
theorem simple_graph.walk.edges_take_until_subset {V : Type u} {G : simple_graph V} [decidable_eq V] {u v w : V} (p : G.walk v w) (h : u ∈ p.support) :
(p.take_until u h).edges ⊆ p.edges
source
theorem simple_graph.walk.edges_drop_until_subset {V : Type u} {G : simple_graph V} [decidable_eq V] {u v w : V} (p : G.walk v w) (h : u ∈ p.support) :
(p.drop_until u h).edges ⊆ p.edges
source
theorem simple_graph.walk.length_take_until_le {V : Type u} {G : simple_graph V} [decidable_eq V] {u v w : V} (p : G.walk v w) (h : u ∈ p.support) :
(p.take_until u h).length ≤ p.length
source
theorem simple_graph.walk.length_drop_until_le {V : Type u} {G : simple_graph V} [decidable_eq V] {u v w : V} (p : G.walk v w) (h : u ∈ p.support) :
(p.drop_until u h).length ≤ p.length
source
@[protected]
theorem simple_graph.walk.is_trail.take_until {V : Type u} {G : simple_graph V} [decidable_eq V] {u v w : V} {p : G.walk v w} (hc : p.is_trail) (h : u ∈ p.support) :
(p.take_until u h).is_trail
source
@[protected]
theorem simple_graph.walk.is_trail.drop_until {V : Type u} {G : simple_graph V} [decidable_eq V] {u v w : V} {p : G.walk v w} (hc : p.is_trail) (h : u ∈ p.support) :
(p.drop_until u h).is_trail
source
@[protected]
theorem simple_graph.walk.is_path.take_until {V : Type u} {G : simple_graph V} [decidable_eq V] {u v w : V} {p : G.walk v w} (hc : p.is_path) (h : u ∈ p.support) :
(p.take_until u h).is_path
source
@[protected]
theorem simple_graph.walk.is_path.drop_until {V : Type u} {G : simple_graph V} [decidable_eq V] {u v w : V} (p : G.walk v w) (hc : p.is_path) (h : u ∈ p.support) :
(p.drop_until u h).is_path
source
def simple_graph.walk.rotate {V : Type u} {G : simple_graph V} [decidable_eq V] {u v : V} (c : G.walk v v) (h : u ∈ c.support) :
G.walk u u

Rotate a loop walk such that it is centered at the given vertex.

Equations
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@[simp]
theorem simple_graph.walk.support_rotate {V : Type u} {G : simple_graph V} [decidable_eq V] {u v : V} (c : G.walk v v) (h : u ∈ c.support) :
(c.rotate h).support.tail ~r c.support.tail
source
theorem simple_graph.walk.rotate_darts {V : Type u} {G : simple_graph V} [decidable_eq V] {u v : V} (c : G.walk v v) (h : u ∈ c.support) :
(c.rotate h).darts ~r c.darts
source
theorem simple_graph.walk.rotate_edges {V : Type u} {G : simple_graph V} [decidable_eq V] {u v : V} (c : G.walk v v) (h : u ∈ c.support) :
(c.rotate h).edges ~r c.edges
source
@[protected]
theorem simple_graph.walk.is_trail.rotate {V : Type u} {G : simple_graph V} [decidable_eq V] {u v : V} {c : G.walk v v} (hc : c.is_trail) (h : u ∈ c.support) :
(c.rotate h).is_trail
source
@[protected]
theorem simple_graph.walk.is_circuit.rotate {V : Type u} {G : simple_graph V} [decidable_eq V] {u v : V} {c : G.walk v v} (hc : c.is_circuit) (h : u ∈ c.support) :
(c.rotate h).is_circuit
source
@[protected]
theorem simple_graph.walk.is_cycle.rotate {V : Type u} {G : simple_graph V} [decidable_eq V] {u v : V} {c : G.walk v v} (hc : c.is_cycle) (h : u ∈ c.support) :
(c.rotate h).is_cycle

Type of paths #

source
@[reducible]
def simple_graph.path {V : Type u} (G : simple_graph V) (u v : V) :
Type u

The type for paths between two vertices.

source
@[protected, simp]
theorem simple_graph.path.is_path {V : Type u} {G : simple_graph V} {u v : V} (p : G.path u v) :
↑p.is_path
source
@[protected, simp]
theorem simple_graph.path.is_trail {V : Type u} {G : simple_graph V} {u v : V} (p : G.path u v) :
↑p.is_trail
source
@[simp]
theorem simple_graph.path.nil_coe {V : Type u} {G : simple_graph V} {u : V} :
↑simple_graph.path.nil = simple_graph.walk.nil
source
@[protected, refl]
def simple_graph.path.nil {V : Type u} {G : simple_graph V} {u : V} :
G.path u u

The length-0 path at a vertex.

Equations
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def simple_graph.path.singleton {V : Type u} {G : simple_graph V} {u v : V} (h : G.adj u v) :
G.path u v

The length-1 path between a pair of adjacent vertices.

Equations
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@[simp]
theorem simple_graph.path.singleton_coe {V : Type u} {G : simple_graph V} {u v : V} (h : G.adj u v) :
↑(simple_graph.path.singleton h) = simple_graph.walk.cons h simple_graph.walk.nil
source
theorem simple_graph.path.mk_mem_edges_singleton {V : Type u} {G : simple_graph V} {u v : V} (h : G.adj u v) :
⟦(u, v)⟧ ∈ ↑(simple_graph.path.singleton h).edges
source
@[simp]
theorem simple_graph.path.reverse_coe {V : Type u} {G : simple_graph V} {u v : V} (p : G.path u v) :
↑(p.reverse) = ↑p.reverse
source
@[symm]
def simple_graph.path.reverse {V : Type u} {G : simple_graph V} {u v : V} (p : G.path u v) :
G.path v u

The reverse of a path is another path. See also simple_graph.walk.reverse.

Equations
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theorem simple_graph.path.count_support_eq_one {V : Type u} {G : simple_graph V} [decidable_eq V] {u v w : V} {p : G.path u v} (hw : w ∈ ↑p.support) :
list.count w ↑p.support = 1
source
theorem simple_graph.path.count_edges_eq_one {V : Type u} {G : simple_graph V} [decidable_eq V] {u v : V} {p : G.path u v} (e : sym2 V) (hw : e ∈ ↑p.edges) :
list.count e ↑p.edges = 1
source
@[simp]
theorem simple_graph.path.nodup_support {V : Type u} {G : simple_graph V} {u v : V} (p : G.path u v) :
↑p.support.nodup
source
theorem simple_graph.path.loop_eq {V : Type u} {G : simple_graph V} {v : V} (p : G.path v v) :
p = simple_graph.path.nil
source
theorem simple_graph.path.not_mem_edges_of_loop {V : Type u} {G : simple_graph V} {v : V} {e : sym2 V} {p : G.path v v} :
e ∉ ↑p.edges
source
theorem simple_graph.path.cons_is_cycle {V : Type u} {G : simple_graph V} {u v : V} (p : G.path v u) (h : G.adj u v) (he : ⟦(u, v)⟧ ∉ ↑p.edges) :
(simple_graph.walk.cons h ↑p).is_cycle

Walks to paths #

source
def simple_graph.walk.bypass {V : Type u} {G : simple_graph V} [decidable_eq V] {u v : V} :
G.walk u v → G.walk u v

Given a walk, produces a walk from it by bypassing subwalks between repeated vertices. The result is a path, as shown in simple_graph.walk.bypass_is_path. This is packaged up in simple_graph.walk.to_path.

Equations
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theorem simple_graph.walk.bypass_is_path {V : Type u} {G : simple_graph V} [decidable_eq V] {u v : V} (p : G.walk u v) :
p.bypass.is_path
source
theorem simple_graph.walk.length_bypass_le {V : Type u} {G : simple_graph V} [decidable_eq V] {u v : V} (p : G.walk u v) :
p.bypass.length ≤ p.length
source
def simple_graph.walk.to_path {V : Type u} {G : simple_graph V} [decidable_eq V] {u v : V} (p : G.walk u v) :
G.path u v

Given a walk, produces a path with the same endpoints using simple_graph.walk.bypass.

Equations
source
theorem simple_graph.walk.support_bypass_subset {V : Type u} {G : simple_graph V} [decidable_eq V] {u v : V} (p : G.walk u v) :
p.bypass.support ⊆ p.support
source
theorem simple_graph.walk.support_to_path_subset {V : Type u} {G : simple_graph V} [decidable_eq V] {u v : V} (p : G.walk u v) :
↑(p.to_path).support ⊆ p.support
source
theorem simple_graph.walk.darts_bypass_subset {V : Type u} {G : simple_graph V} [decidable_eq V] {u v : V} (p : G.walk u v) :
p.bypass.darts ⊆ p.darts
source
theorem simple_graph.walk.edges_bypass_subset {V : Type u} {G : simple_graph V} [decidable_eq V] {u v : V} (p : G.walk u v) :
p.bypass.edges ⊆ p.edges
source
theorem simple_graph.walk.darts_to_path_subset {V : Type u} {G : simple_graph V} [decidable_eq V] {u v : V} (p : G.walk u v) :
↑(p.to_path).darts ⊆ p.darts
source
theorem simple_graph.walk.edges_to_path_subset {V : Type u} {G : simple_graph V} [decidable_eq V] {u v : V} (p : G.walk u v) :
↑(p.to_path).edges ⊆ p.edges

Mapping paths #

source
@[protected]
def simple_graph.walk.map {V : Type u} {V' : Type v} {G : simple_graph V} {G' : simple_graph V'} (f : G →g G') {u v : V} :
G.walk u v → G'.walk (⇑f u) (⇑f v)

Given a graph homomorphism, map walks to walks.

Equations
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@[simp]
theorem simple_graph.walk.map_nil {V : Type u} {V' : Type v} {G : simple_graph V} {G' : simple_graph V'} (f : G →g G') {u : V} :
simple_graph.walk.map f simple_graph.walk.nil = simple_graph.walk.nil
source
@[simp]
theorem simple_graph.walk.map_cons {V : Type u} {V' : Type v} {G : simple_graph V} {G' : simple_graph V'} (f : G →g G') {u v : V} (p : G.walk u v) {w : V} (h : G.adj w u) :
simple_graph.walk.map f (simple_graph.walk.cons h p) = simple_graph.walk.cons _ (simple_graph.walk.map f p)
source
@[simp]
theorem simple_graph.walk.length_map {V : Type u} {V' : Type v} {G : simple_graph V} {G' : simple_graph V'} (f : G →g G') {u v : V} (p : G.walk u v) :
(simple_graph.walk.map f p).length = p.length
source
theorem simple_graph.walk.map_append {V : Type u} {V' : Type v} {G : simple_graph V} {G' : simple_graph V'} (f : G →g G') {u v w : V} (p : G.walk u v) (q : G.walk v w) :
simple_graph.walk.map f (p.append q) = (simple_graph.walk.map f p).append (simple_graph.walk.map f q)
source
@[simp]
theorem simple_graph.walk.reverse_map {V : Type u} {V' : Type v} {G : simple_graph V} {G' : simple_graph V'} (f : G →g G') {u v : V} (p : G.walk u v) :
(simple_graph.walk.map f p).reverse = simple_graph.walk.map f p.reverse
source
@[simp]
theorem simple_graph.walk.support_map {V : Type u} {V' : Type v} {G : simple_graph V} {G' : simple_graph V'} (f : G →g G') {u v : V} (p : G.walk u v) :
(simple_graph.walk.map f p).support = list.map ⇑f p.support
source
@[simp]
theorem simple_graph.walk.darts_map {V : Type u} {V' : Type v} {G : simple_graph V} {G' : simple_graph V'} (f : G →g G') {u v : V} (p : G.walk u v) :
(simple_graph.walk.map f p).darts = list.map f.map_dart p.darts
source
@[simp]
theorem simple_graph.walk.edges_map {V : Type u} {V' : Type v} {G : simple_graph V} {G' : simple_graph V'} (f : G →g G') {u v : V} (p : G.walk u v) :
(simple_graph.walk.map f p).edges = list.map (sym2.map ⇑f) p.edges
source
theorem simple_graph.walk.map_is_path_of_injective {V : Type u} {V' : Type v} {G : simple_graph V} {G' : simple_graph V'} {f : G →g G'} {u v : V} {p : G.walk u v} (hinj : function.injective ⇑f) (hp : p.is_path) :
(simple_graph.walk.map f p).is_path
source
@[protected]
theorem simple_graph.walk.is_path.of_map {V : Type u} {V' : Type v} {G : simple_graph V} {G' : simple_graph V'} {u v : V} {p : G.walk u v} {f : G →g G'} (hp : (simple_graph.walk.map f p).is_path) :
p.is_path
source
theorem simple_graph.walk.map_is_path_iff_of_injective {V : Type u} {V' : Type v} {G : simple_graph V} {G' : simple_graph V'} {f : G →g G'} {u v : V} {p : G.walk u v} (hinj : function.injective ⇑f) :
(simple_graph.walk.map f p).is_path ↔ p.is_path
source
theorem simple_graph.walk.map_injective_of_injective {V : Type u} {V' : Type v} {G : simple_graph V} {G' : simple_graph V'} {f : G →g G'} (hinj : function.injective ⇑f) (u v : V) :
function.injective (simple_graph.walk.map f)
source
@[simp]
theorem simple_graph.path.map_coe {V : Type u} {V' : Type v} {G : simple_graph V} {G' : simple_graph V'} (f : G →g G') (hinj : function.injective ⇑f) {u v : V} (p : G.path u v) :
↑(simple_graph.path.map f hinj p) = simple_graph.walk.map f ↑p
source
@[protected]
def simple_graph.path.map {V : Type u} {V' : Type v} {G : simple_graph V} {G' : simple_graph V'} (f : G →g G') (hinj : function.injective ⇑f) {u v : V} (p : G.path u v) :
G'.path (⇑f u) (⇑f v)

Given an injective graph homomorphism, map paths to paths.

Equations
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theorem simple_graph.path.map_injective {V : Type u} {V' : Type v} {G : simple_graph V} {G' : simple_graph V'} {f : G →g G'} (hinj : function.injective ⇑f) (u v : V) :
function.injective (simple_graph.path.map f hinj)
source
@[protected]
def simple_graph.path.map_embedding {V : Type u} {V' : Type v} {G : simple_graph V} {G' : simple_graph V'} (f : G ↪g G') {u v : V} (p : G.path u v) :
G'.path (⇑f u) (⇑f v)

Given a graph embedding, map paths to paths.

Equations
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@[simp]
theorem simple_graph.path.map_embedding_coe {V : Type u} {V' : Type v} {G : simple_graph V} {G' : simple_graph V'} (f : G ↪g G') {u v : V} (p : G.path u v) :
↑(simple_graph.path.map_embedding f p) = simple_graph.walk.map f.to_hom ↑p
source
theorem simple_graph.path.map_embedding_injective {V : Type u} {V' : Type v} {G : simple_graph V} {G' : simple_graph V'} (f : G ↪g G') (u v : V) :
function.injective (simple_graph.path.map_embedding f)

Deleting edges #

source
@[simp]
def simple_graph.walk.to_delete_edges {V : Type u} {G : simple_graph V} (s : set (sym2 V)) {v w : V} (p : G.walk v w) (hp : ∀ (e : sym2 V), e ∈ p.edges → e ∉ s) :
(G.delete_edges s).walk v w

Given a walk that avoids a set of edges, produce a walk in the graph with those edges deleted.

Equations
source
@[reducible]
def simple_graph.walk.to_delete_edge {V : Type u} {G : simple_graph V} {v w : V} (e : sym2 V) (p : G.walk v w) (hp : e ∉ p.edges) :
(G.delete_edges {e}).walk v w

Given a walk that avoids an edge, create a walk in the subgraph with that edge deleted. This is an abbreviation for simple_graph.walk.to_delete_edges.

source
@[simp]
theorem simple_graph.walk.map_to_delete_edges_eq {V : Type u} {G : simple_graph V} (s : set (sym2 V)) {v w : V} {p : G.walk v w} (hp : ∀ (e : sym2 V), e ∈ p.edges → e ∉ s) :
simple_graph.walk.map (simple_graph.hom.map_spanning_subgraphs _) (simple_graph.walk.to_delete_edges s p hp) = p
source
theorem simple_graph.walk.is_path.to_delete_edges {V : Type u} {G : simple_graph V} (s : set (sym2 V)) {v w : V} {p : G.walk v w} (h : p.is_path) (hp : ∀ (e : sym2 V), e ∈ p.edges → e ∉ s) :
(simple_graph.walk.to_delete_edges s p hp).is_path

reachable and connected #

source
def simple_graph.reachable {V : Type u} (G : simple_graph V) (u v : V) :
Prop

Two vertices are reachable if there is a walk between them. This is equivalent to relation.refl_trans_gen of G.adj. See simple_graph.reachable_iff_refl_trans_gen.

Equations
source
theorem simple_graph.reachable_iff_nonempty_univ {V : Type u} {G : simple_graph V} {u v : V} :
G.reachable u v ↔ set.univ.nonempty
source
@[protected]
theorem simple_graph.reachable.elim {V : Type u} {G : simple_graph V} {p : Prop} {u v : V} (h : G.reachable u v) (hp : G.walk u v → p) :
p
source
@[protected]
theorem simple_graph.reachable.elim_path {V : Type u} {G : simple_graph V} {p : Prop} {u v : V} (h : G.reachable u v) (hp : G.path u v → p) :
p
source
@[protected, refl]
theorem simple_graph.reachable.refl {V : Type u} {G : simple_graph V} (u : V) :
G.reachable u u
source
@[protected]
theorem simple_graph.reachable.rfl {V : Type u} {G : simple_graph V} {u : V} :
G.reachable u u
source
@[protected, symm]
theorem simple_graph.reachable.symm {V : Type u} {G : simple_graph V} {u v : V} (huv : G.reachable u v) :
G.reachable v u
source
@[protected, trans]
theorem simple_graph.reachable.trans {V : Type u} {G : simple_graph V} {u v w : V} (huv : G.reachable u v) (hvw : G.reachable v w) :
G.reachable u w
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theorem simple_graph.reachable_iff_refl_trans_gen {V : Type u} {G : simple_graph V} (u v : V) :
G.reachable u v ↔ relation.refl_trans_gen G.adj u v
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theorem simple_graph.reachable_is_equivalence {V : Type u} (G : simple_graph V) :
equivalence G.reachable
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def simple_graph.reachable_setoid {V : Type u} (G : simple_graph V) :
setoid V

The equivalence relation on vertices given by simple_graph.reachable.

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def simple_graph.preconnected {V : Type u} (G : simple_graph V) :
Prop

A graph is preconnected if every pair of vertices is reachable from one another.

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theorem simple_graph.preconnected.map {V : Type u} {V' : Type v} {G : simple_graph V} {H : simple_graph V'} (f : G →g H) (hf : function.surjective ⇑f) (hG : G.preconnected) :
H.preconnected
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theorem simple_graph.iso.preconnected_iff {V : Type u} {V' : Type v} {G : simple_graph V} {H : simple_graph V'} (e : G ≃g H) :
G.preconnected ↔ H.preconnected
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theorem simple_graph.connected_iff {V : Type u} (G : simple_graph V) :
G.connected ↔ G.preconnected ∧ nonempty V
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structure simple_graph.connected {V : Type u} (G : simple_graph V) :
Prop

A graph is connected if it's preconnected and contains at least one vertex. This follows the convention observed by mathlib that something is connected iff it has exactly one connected component.

There is a has_coe_to_fun instance so that h u v can be used instead of h.preconnected u v.

Instances for simple_graph.connected
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@[protected, instance]
def simple_graph.connected.has_coe_to_fun {V : Type u} (G : simple_graph V) :
has_coe_to_fun G.connected (λ (_x : G.connected), ∀ (u v : V), G.reachable u v)
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theorem simple_graph.connected.map {V : Type u} {V' : Type v} {G : simple_graph V} {H : simple_graph V'} (f : G →g H) (hf : function.surjective ⇑f) (hG : G.connected) :
H.connected
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theorem simple_graph.iso.connected_iff {V : Type u} {V' : Type v} {G : simple_graph V} {H : simple_graph V'} (e : G ≃g H) :
G.connected ↔ H.connected
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def simple_graph.connected_component {V : Type u} (G : simple_graph V) :
Type u

The quotient of V by the simple_graph.reachable relation gives the connected components of a graph.

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Instances for simple_graph.connected_component
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def simple_graph.connected_component_mk {V : Type u} (G : simple_graph V) (v : V) :
G.connected_component

Gives the connected component containing a particular vertex.

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@[simp]
theorem simple_graph.connected_component.inhabited_default {V : Type u} (G : simple_graph V) [inhabited V] :
inhabited.default = G.connected_component_mk inhabited.default
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@[protected, instance]
def simple_graph.connected_component.inhabited {V : Type u} (G : simple_graph V) [inhabited V] :
inhabited G.connected_component
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@[protected]
theorem simple_graph.connected_component.ind {V : Type u} {G : simple_graph V} {β : G.connected_component → Prop} (h : ∀ (v : V), β (G.connected_component_mk v)) (c : G.connected_component) :
β c
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@[protected]
theorem simple_graph.connected_component.ind₂ {V : Type u} {G : simple_graph V} {β : G.connected_component → G.connected_component → Prop} (h : ∀ (v w : V), β (G.connected_component_mk v) (G.connected_component_mk w)) (c d : G.connected_component) :
β c d
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@[protected]
theorem simple_graph.connected_component.sound {V : Type u} {G : simple_graph V} {v w : V} :
G.reachable v w → G.connected_component_mk v = G.connected_component_mk w
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@[protected]
theorem simple_graph.connected_component.exact {V : Type u} {G : simple_graph V} {v w : V} :
G.connected_component_mk v = G.connected_component_mk w → G.reachable v w
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@[protected, simp]
theorem simple_graph.connected_component.eq {V : Type u} {G : simple_graph V} {v w : V} :
G.connected_component_mk v = G.connected_component_mk w ↔ G.reachable v w
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@[protected]
def simple_graph.connected_component.lift {V : Type u} {G : simple_graph V} {β : Sort u_1} (f : V → β) (h : ∀ (v w : V) (p : G.walk v w), p.is_path → f v = f w) :
G.connected_component → β

The connected_component specialization of quot.lift. Provides the stronger assumption that the vertices are connected by a path.

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@[protected, simp]
theorem simple_graph.connected_component.lift_mk {V : Type u} {G : simple_graph V} {β : Sort u_1} {f : V → β} {h : ∀ (v w : V) (p : G.walk v w), p.is_path → f v = f w} {v : V} :
simple_graph.connected_component.lift f h (G.connected_component_mk v) = f v
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@[protected]
theorem simple_graph.connected_component.exists {V : Type u} {G : simple_graph V} {p : G.connected_component → Prop} :
(∃ (c : G.connected_component), p c) ↔ ∃ (v : V), p (G.connected_component_mk v)
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@[protected]
theorem simple_graph.connected_component.forall {V : Type u} {G : simple_graph V} {p : G.connected_component → Prop} :
(∀ (c : G.connected_component), p c) ↔ ∀ (v : V), p (G.connected_component_mk v)
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theorem simple_graph.preconnected.subsingleton_connected_component {V : Type u} {G : simple_graph V} (h : G.preconnected) :
subsingleton G.connected_component
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@[reducible]
def simple_graph.subgraph.connected {V : Type u} {G : simple_graph V} (H : G.subgraph) :
Prop

A subgraph is connected if it is connected as a simple graph.

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theorem simple_graph.preconnected.set_univ_walk_nonempty {V : Type u} {G : simple_graph V} (hconn : G.preconnected) (u v : V) :
set.univ.nonempty
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theorem simple_graph.connected.set_univ_walk_nonempty {V : Type u} {G : simple_graph V} (hconn : G.connected) (u v : V) :
set.univ.nonempty

Walks of a given length #

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theorem simple_graph.set_walk_self_length_zero_eq {V : Type u} {G : simple_graph V} (u : V) :
{p : G.walk u u | p.length = 0} = {simple_graph.walk.nil}
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theorem simple_graph.set_walk_length_zero_eq_of_ne {V : Type u} {G : simple_graph V} {u v : V} (h : u ≠ v) :
{p : G.walk u v | p.length = 0} = ∅
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theorem simple_graph.set_walk_length_succ_eq {V : Type u} {G : simple_graph V} (u v : V) (n : ℕ) :
{p : G.walk u v | p.length = n.succ} = ⋃ (w : V) (h : G.adj u w), simple_graph.walk.cons h '' {p' : G.walk w v | p'.length = n}
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def simple_graph.finset_walk_length {V : Type u} (G : simple_graph V) [fintype V] [decidable_rel G.adj] [decidable_eq V] (n : ℕ) (u v : V) :
finset (G.walk u v)

The finset of length-n walks from u to v. This is used to give {p : G.walk u v | p.length = n} a fintype instance, and it can also be useful as a recursive description of this set when V is finite.

See simple_graph.coe_finset_walk_length_eq for the relationship between this finset and the set of length-n walks.

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theorem simple_graph.coe_finset_walk_length_eq {V : Type u} (G : simple_graph V) [fintype V] [decidable_rel G.adj] [decidable_eq V] (n : ℕ) (u v : V) :
↑(G.finset_walk_length n u v) = {p : G.walk u v | p.length = n}
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theorem simple_graph.walk.length_eq_of_mem_finset_walk_length {V : Type u} {G : simple_graph V} [fintype V] [decidable_rel G.adj] [decidable_eq V] {n : ℕ} {u v : V} (p : G.walk u v) :
p ∈ G.finset_walk_length n u v → p.length = n
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@[protected, instance]
def simple_graph.fintype_set_walk_length {V : Type u} (G : simple_graph V) [fintype V] [decidable_rel G.adj] [decidable_eq V] (u v : V) (n : ℕ) :
fintype ↥{p : G.walk u v | p.length = n}
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theorem simple_graph.set_walk_length_to_finset_eq {V : Type u} (G : simple_graph V) [fintype V] [decidable_rel G.adj] [decidable_eq V] (n : ℕ) (u v : V) :
{p : G.walk u v | p.length = n}.to_finset = G.finset_walk_length n u v
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theorem simple_graph.card_set_walk_length_eq {V : Type u} (G : simple_graph V) [fintype V] [decidable_rel G.adj] [decidable_eq V] (u v : V) (n : ℕ) :
fintype.card ↥{p : G.walk u v | p.length = n} = (G.finset_walk_length n u v).card