 Planar graph

A planar graph admits at least one graph drawing in which edges meet only at common endpoints. This is a property of the abstract graph, not a requirement on every graph drawing. A plane graph specifies one such drawing.^[ref]

For example, the complete graph $K_4$ joins every pair of four vertices, giving six edges. Put its vertices at the corners of a square and its diagonals cross. Instead put $A,B,C$ at the corners of a triangle and $D$ inside it: the same six edges $AB,AC,BC,AD,BD,CD$ now meet only at their endpoints. The first graph drawing obscures planarity; the second proves it. In particular, its crossing number is zero.

The distinction matters whenever connections must fit on one surface without intersecting. A failed graph drawing may only need rearrangement, whereas nonplanarity by edge counts proves that every rearrangement fails.

A **planar graph** is a type of graph that can be embedded in the plane, meaning that it can be drawn on a flat surface such that its edges intersect only at their endpoints (vertices) and do not cross each other. In other words, a graph is planar if it can be represented in such a way that no two edges overlap except at their endpoints.