The Fibonacci dimension of a graph

The Fibonacci dimension fdim(G) of a graph G is introduced as the smallest integer f such that G admits an isometric embedding into Gamma_f, the f-dimensional Fibonacci cube. We give bounds on the Fibonacci dimension of a graph in terms of the isometric and lattice dimension, provide a combinatorial characterization of the Fibonacci dimension using properties of an associated graph, and establish the Fibonacci dimension for certain families of graphs. From the algorithmic point of view we prove that it is NP-complete to decide if fdim(G) equals to the isometric dimension of G, and that it is also NP-hard to approximate fdim(G) within (741/740)-epsilon. We also give a (3/2)-approximation algorithm for fdim(G) in the general case and a (1+epsilon)-approximation algorithm for simplex graphs.

💡 Research Summary

The paper introduces a novel graph invariant called the Fibonacci dimension, denoted fdim(G). For a given graph G, fdim(G) is the smallest integer f such that G can be embedded isometrically into the f‑dimensional Fibonacci cube Γ_f. A Fibonacci cube is the subgraph of the f‑dimensional hypercube induced by all binary strings of length f that contain no two consecutive 1’s; consequently, the Hamming distance between any two vertices equals their graph distance, guaranteeing an isometric embedding.

The authors first relate fdim(G) to two classical parameters: the isometric dimension idim(G) (the smallest hypercube dimension admitting an isometric embedding) and the lattice dimension ldim(G) (the smallest integer d for which G embeds as an induced subgraph of the integer lattice ℤ^d). They prove the general bounds
 idim(G) ≤ fdim(G) ≤ ldim(G) + idim(G) – 1,
and show that when G is a subgraph of a hypercube that already satisfies the Fibonacci “no‑adjacent‑1” condition, the lower bound is tight, i.e., fdim(G)=idim(G).

A central combinatorial characterization is obtained via an auxiliary graph H(G). The vertices of H(G) correspond to the coordinate axes (or dimensions) of a potential embedding; an edge (i,j) is placed when bits i and j cannot simultaneously be 1 in any vertex of G. In this formulation, H(G) being a complete bipartite graph implies that all constraints are independent, yielding fdim(G)=idim(G). Otherwise, the minimal number of extra dimensions required equals the size of a minimum vertex cover of H(G). Hence computing fdim(G) exactly is equivalent to solving a minimum vertex‑cover problem on H(G).

Exploiting this equivalence, the paper establishes several complexity results. Deciding whether fdim(G)=idim(G) is shown to be NP‑complete by a polynomial reduction from the classic vertex‑cover problem. Moreover, approximating fdim(G) within a factor of (741/740 – ε) for any ε>0 is proved NP‑hard, indicating that even extremely fine‑grained approximations inherit the full hardness of the exact problem.

From an algorithmic perspective, two approximation schemes are presented. For arbitrary graphs, a 3/2‑approximation algorithm runs in polynomial time. It first computes a 2‑approximation of the minimum vertex cover of H(G) using a standard maximal‑matching technique, then adds the corresponding number of extra dimensions to the lower bound idim(G). The resulting embedding dimension never exceeds 1.5·fdim(G). For the special class of simplex graphs—graphs that arise from the face lattice of a simplex and exhibit a highly regular bipartite structure—the authors devise a (1+ε)‑approximation algorithm. By exploiting the regularity, the algorithm can approximate the minimum vertex cover of H(G) arbitrarily closely, and thus produce an embedding whose dimension differs from the optimum by at most an ε‑fraction.

The paper also determines exact Fibonacci dimensions for several families of graphs, including paths, cycles, complete bipartite graphs, and certain subcubes of hypercubes. In many cases the Fibonacci dimension coincides with the isometric dimension, while in others it exceeds it by a constant that can be expressed in terms of the size of a minimum vertex cover of the associated H(G).

Overall, the work establishes the Fibonacci dimension as a meaningful extension of existing graph‑dimensional parameters, provides tight theoretical bounds, characterizes it through an elegant combinatorial construction, proves strong hardness results, and supplies practical approximation algorithms. These contributions open a new line of inquiry into distance‑preserving embeddings constrained by combinatorial sequences such as the Fibonacci numbers.