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Lax68.StraightLineDrawings

Straight-line graph drawings

concepts/Lax68/StraightLineDrawings.lean · lax-68

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    Definition

    A straight-line drawing assigns distinct points of the real plane to the vertices of a simple graph and draws every edge as the segment between its endpoints. No vertex lies inside an edge and disjoint edges do not meet. This is the geometric drawing certificate used by the planar-graph concept.

    Lean source view on GitHub

    1import Mathlib.Analysis.Convex.Segment
    2import Mathlib.Combinatorics.SimpleGraph.Basic
    3import Mathlib.Data.Real.Basic
    4
    5/-!
    6---
    7title: Straight-line graph drawings
    8type: definition
    9---
    10A straight-line drawing assigns distinct points of the real plane to the
    11vertices of a simple graph and draws every edge as the segment between its
    12endpoints. No vertex lies inside an edge and disjoint edges do not meet.
    13This is the geometric drawing certificate used by the planar-graph concept.
    14-/
    15
    16set_option autoImplicit false
    17
    18namespace Lax68.StraightLineDrawings
    19
    20abbrev Point := ℝ × ℝ
    21
    22structure StraightLineDrawing {V : Type*} (G : SimpleGraph V) where
    23 point : V → Point
    24 injective : Function.Injective point
    25 noVertexOnEdge :
    26 ∀ {a b c : V},
    27 G.Adj a b →
    28 c ≠ a →
    29 c ≠ b →
    30 point c ∉ segment ℝ (point a) (point b)
    31 disjointEdges :
    32 ∀ {a b c d : V},
    33 G.Adj a b →
    34 G.Adj c d →
    35 Disjoint ({a, b} : Set V) ({c, d} : Set V) →
    36 Disjoint
    37 (segment ℝ (point a) (point b))
    38 (segment ℝ (point c) (point d))
    39
    40def HasStraightLineDrawing {V : Type*} (G : SimpleGraph V) : Prop :=
    41 Nonempty (StraightLineDrawing G)
    42
    43end Lax68.StraightLineDrawings
    44

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    Used by

    From Mathlib

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