---
tag: HFJCAPWWD4
kind: concept
name: Geometric graph
msc: [05C62, 52C10, 05C10]
sources:
  - url: https://en.wikipedia.org/wiki/Geometric_graph_theory
---

## Summary

A geometric graph is a [[graph|00C0BR7D7H]] drawn in the plane with its vertices as points and its edges as straight line segments, which may cross.
Geometric graph theory asks how the positions of the points constrain the graph.
A typical question is how many edges force two edges to cross.
A geometric graph without crossings is a straight-line drawing of a [[planar graph|0NC9R64TXP]].
The subject sits between graph theory and [[discrete geometry|Q0RW86P85F]].

## Explanations

### L1 human

Mark some dots on paper and join some pairs of dots with a ruler.
The result is a geometric graph: the dots are vertices and the straight strokes are edges.
For example, the four corners of a square joined by its four sides and two diagonals give a geometric graph of the complete graph $K_4$ with one crossing.
Placing one corner inside the triangle of the other three gives a drawing of $K_4$ with no crossing.
So the same abstract graph can have different geometric drawings with different properties.