Matchstick graph

In geometric graph theory, a branch of mathematics, a matchstick graph is a graph that can be drawn in the plane in such a way that its edges are line segments with length one that do not cross each other. That is, it is a graph that has an embedding which is simultaneously a unit distance graph and a plane graph. Informally, matchstick graphs can be made by placing noncrossing matchsticks on a flat surface, hence the name.

Matchstick graphs have also been called planar unit-distance graphs. However, to be a matchstick graph, the unit distance embedding must be plane; it is not enough for the graph to be planar. Some graphs have both non-crossing embeddings with non-unit vertex distances and unit-distance embeddings with crossings; these are planar unit-distance graphs but are not matchstick graphs. An example is the Dürer graph.