De Bruijn graph

In graph theory, an n-dimensional De Bruijn graph of m symbols is a directed graph representing overlaps between sequences of symbols. It has mⁿ vertices, consisting of all possible length-n sequences of the given symbols; the same symbol may appear multiple times in a sequence. For a set of m symbols S = {s₁, …, s_m}, the set of vertices is:

${\displaystyle V=S^{n}=\{(s_{1},\dots ,s_{1},s_{1}),(s_{1},\dots ,s_{1},s_{2}),\dots ,(s_{1},\dots ,s_{1},s_{m}),(s_{1},\dots ,s_{2},s_{1}),\dots ,(s_{m},\dots ,s_{m},s_{m})\}.}$

If one of the vertices can be expressed as another vertex by shifting all its symbols by one place to the left and adding a new symbol at the end of this vertex, then the latter has a directed edge to the former vertex. Thus the set of arcs (that is, directed edges) is

${\displaystyle E=\{((t_{1},t_{2},\dots ,t_{n}),(t_{2},\dots ,t_{n},s_{j})):t_{i}\in S,1\leq i\leq n,1\leq j\leq m\}.}$

Although De Bruijn graphs are named after Nicolaas Govert de Bruijn, they were invented independently by both de Bruijn and I. J. Good. Much earlier, Camille Flye Sainte-Marie implicitly used their properties.