FormalConjecturesForMathlib.Combinatorics.SimpleGraph.SubgraphIsomorphism

/-
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.
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    https://www.apache.org/licenses/LICENSE-2.0

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-/
module

public import Mathlib.Combinatorics.SimpleGraph.Maps
public import Mathlib.Data.Set.Card

@[expose] public sectionnamespace SimpleGraphvariable {V : Type*}

G is a unique subgraph of H if there is exactly one subgraph of H isomorphic to G. Subgraphs of H are taken in the spanning sense: elements G' ≤ H of the lattice SimpleGraph V, i.e. subsets of the edges of H (not necessarily induced).

def IsUniqueSubgraph (G H : SimpleGraph V) : Prop :=
  ∃! G' : SimpleGraph V, G' ≤ H ∧ Nonempty (G.Iso G')

The number of distinct unique subgraphs of H: the number of isomorphism classes of graphs that occur exactly once as a subgraph of H. Each such class contains exactly one subgraph G ≤ H (uniqueness forbids a second isomorphic copy), so counting those representatives counts the classes.

noncomputable def uniqueSubgraphCount (H : SimpleGraph V) : ℕ :=
  {G : SimpleGraph V | G ≤ H ∧ IsUniqueSubgraph G H}.ncard

end SimpleGraph