Fibonacci cube

Fibonacci cubes were first explicitly defined in Hsu (1993) in the context of interconnection topologies for connecting parallel or distributed systems.

[2] That is, each vertex in the Fibonacci cube represents a clique in the path complement graph, or equivalently an independent set in the path itself; two Fibonacci cube vertices are adjacent if the cliques or independent sets that they represent differ by the addition or removal of a single element.

The Fibonacci cube is also the graph of a distributive lattice that may be obtained via Birkhoff's representation theorem from a zigzag poset, a partially ordered set defined by an alternating sequence of order relations a < b > c < d > e < f > ...[4] There is also an alternative graph-theoretic description of the same lattice: the independent sets of any bipartite graph may be given a partial order in which one independent set is less than another if they differ by removing elements from one side of the bipartition and adding elements to the other side of the bipartition; with this order, the independent sets form a distributive lattice,[5] and applying this construction to a path graph results in the lattice associated with the Fibonacci cube.

Thus, e.g., in the Fibonacci cube of order 4, the sequence constructed in this way is (0100-0101-0001-0000-0010)-(1010-1000-1001), where the parentheses demark the subsequences within the two subgraphs of the partition.

[9] Taranenko & Vesel (2007) showed that it is possible to test whether a graph is a Fibonacci cube in time near-linear in its size.

[2] Generalized Fibonacci cubes were presented by Hsu & Chung (1993) based on the k-th order Fibonacci numbers, which were later further extended to a larger class of networks called the Linear Recursive Networks by Hsu, Chung & Das (1997) based on more general forms of linear recursions.