/-
Copyright 2026 The Formal Conjectures Authors.
Licensed under the Apache License, Version 2.0 (the "License");
you may not use this file except in compliance with the License.
You may obtain a copy of the License at
https://www.apache.org/licenses/LICENSE-2.0
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-/
module
public import Mathlib.Combinatorics.SimpleGraph.Paths
public import Mathlib.Data.Real.Basic
public import Mathlib.Order.Lattice.Nat@[expose] public sectionnamespace SimpleGraphvariable {α : Type*} [Fintype α] [DecidableEq α]A family of paths covering all vertices without overlaps.
def IsPathCover (G : SimpleGraph α) (P : Finset (Finset α)) : Prop :=
(∀ s1 ∈ P, ∀ s2 ∈ P, s1 ≠ s2 → Disjoint s1 s2) ∧
(Finset.univ ⊆ P.biUnion id) ∧
(∀ s ∈ P, ∃ (u v : α) (p : G.Walk u v), p.IsPath ∧ s = p.support.toFinset)
Minimum size of a path cover of G.
noncomputable def pathCoverNumber (G : SimpleGraph α) : ℕ :=
sInf { k | ∃ P : Finset (Finset α), P.card = k ∧ IsPathCover G P }end SimpleGraph