The Fibonacci dimension of a graph

TL;DR

This paper introduces the Fibonacci dimension of a graph, explores its theoretical properties, bounds, and computational complexity, and provides approximation algorithms for calculating it.

Contribution

It defines the Fibonacci dimension of a graph, characterizes it combinatorially, and analyzes its computational complexity and approximation algorithms.

Findings

Fibonacci dimension bounds in terms of isometric and lattice dimensions

NP-completeness of deciding if Fibonacci dimension equals isometric dimension

Approximation algorithms with specific ratios for Fibonacci dimension

Abstract

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.