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Distance-hereditary graphs

Lax560391.DistanceHereditaryGraphs · concepts/Lax560391/DistanceHereditaryGraphs.lean · lax-560391

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    Natural Language Statement

    Definition

    A graph is distance-hereditary if every connected induced subgraph preserves the distances between its vertices. Extended shortest-path distance is used to distinguish disconnected pairs. Empty and disconnected graphs are allowed.

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    1import Mathlib.Combinatorics.SimpleGraph.Metric
    2
    3/-!
    4---
    5title: Distance-hereditary graphs
    6type: definition
    7---
    8A graph is distance-hereditary if every connected induced subgraph preserves
    9the distances between its vertices. Extended shortest-path distance is used
    10to distinguish disconnected pairs. Empty and disconnected graphs are allowed.
    11-/
    12
    13namespace Lax560391.DistanceHereditaryGraphs
    14
    15/-- Every connected induced subgraph retains the distances of `G`.
    16Extended distance avoids identifying disconnected pairs with distance zero. -/
    17def IsDistanceHereditary {V : Type} (G : SimpleGraph V) : Prop :=
    18 ∀ S : Set V, (G.induce S).Connected →
    19 ∀ u v : S, (G.induce S).edist u v = G.edist u.1 v.1
    20
    21end Lax560391.DistanceHereditaryGraphs
    22
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