Bruijngraph

Bruijngraph is a theoretical construct in graph theory and computational biology that generalizes the concept of a Bruijn graph. Named as a portmanteau of Bruijn and a Dutch orthographic variant, it appears in discussions of alternative representations for k-gram overlaps and sequence reconstruction. The Bruijngraph framework uses a finite alphabet Σ and an integer k ≥ 1 to define a class of directed, labeled multigraphs.

In a typical Bruijngraph of order k, vertices correspond to strings of length k−1 over Σ, and directed edges connect a vertex u to a vertex v whenever the concatenation of u with a single symbol a ∈ Σ yields a string whose last k−1 symbols equal v. Edges are labeled by the appended symbol, and multiple edges between the same pair of vertices may occur if the same transition can be realized by different underlying sequences. This construction mirrors the intuition of De Bruijn graphs while allowing variations that reflect data-specific constraints or modeling preferences.

Variations of the Bruijngraph relax or extend this construction: edge lengths may vary, edge labels can carry richer information, or vertices may be grouped into equivalence classes to reflect symmetry or domain-specific constraints. Bruijngraphs retain many useful properties of De Bruijn graphs, notably the connection between Eulerian paths and sequences that generate all desired substrings, enabling compact representations and reconstruction tasks.

Applications include DNA sequence assembly, text indexing, and network data analysis, where efficient handling of overlapping substrings is advantageous. See also Bruijn graph, De Bruijn graph, Eulerian path. Bruijngraph remains a theoretical construct with potential for specialized modelling, rather than a widely adopted standard in mainstream literature.