This is the Harborth graph, the simplest known planar graph in which all the edges are the same length and every vertex has exactly four neighbours.

Not all such graphs are rigid. This one can squish about a bit.

That diagram is from this paper, which has many such diagrams. They also look at a variation where we choose some number k ≠ 4 and say that every vertex must have either 4 or k neighbours.

https://arxiv.org/pdf/1604.07134.pdf

Here’s one in which each vertex has either four or eleven neighbours. (You can see two stars formed by the two vertices that have 11 neighbours; the others have 4.)

The image of the Harborth graph that I posted was generated by Claudio Rocchini, using coordinates computed by Eberhard Gerbracht.

• The image, and the code used to generate it: https://commons.wikimedia.org/wiki/File:Harborth_graph_vector.svg

• The computation of the coordinates: https://arxiv.org/abs/math/0609360

Here’s Harborth’s picture of his graph, from his article “Match Sticks in the Plane.” in The Lighter Side of Mathematics: Proceedings of the Eugène Strens Memorial Conference of Recreational Mathematics & its History. Calgary, Canada, July 27-August 2, 1986