Unit distance graph

In mathematics, particularly geometric graph theory, a unit distance graph is a graph formed from a collection of points in the Euclidean plane by connecting two points whenever the distance between them is exactly one. To distinguish these graphs from a broader definition that allows some non-adjacent pairs of vertices to be at distance one, they may also be called strict unit distance graphs or faithful unit distance graphs. As a hereditary family of graphs, they can be characterized by forbidden induced subgraphs. The unit distance graphs include the cactus graphs, the matchstick graphs and penny graphs, and the hypercube graphs. The generalized Petersen graphs are non-strict unit distance graphs.

A problem posed by Paul Erdős known as the unit distance problem asks for the maximum possible number of unit-distance pairs determined by ${\displaystyle n}$ points in the Euclidean plane; equivalently, it asks for the maximum number of edges in a unit distance graph on ${\displaystyle n}$ vertices. The best known upper bound is ${\displaystyle O(n^{4/3}).}$ The best known lower bound is ${\displaystyle \Omega (n^{1.014})}$ (for sufficiently large ${\displaystyle n)}$. The number of colors required to color T unit distance graphs is also unknown; in the equivalent Hadwiger–Nelson problem for the plane, some unit distance graphs require five colors, and every unit distance graph can be colored with seven colors. For every algebraic number ${\displaystyle \alpha ,}$ there is a unit distance graph with two vertices that must be at distance ${\displaystyle \alpha .}$ According to the Beckman–Quarles theorem, the only plane transformations that preserve all unit distance graphs are the isometries.

It is possible to construct a unit distance graph efficiently, given its points. Finding all unit distances has applications in pattern matching, where it can be a first step in finding congruent copies of larger patterns. However, determining whether a given graph can be represented as a unit distance graph is NP-hard, and more specifically complete for the existential theory of the reals.