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A partial cube is a graph G that can be isometrically embedded into a hypercube \( Q_k \) , with the minimum of such k called the isometric dimension, \(\textrm{idim}(G)\) , of G. A Fibonacci cube \( \varGamma _k \) excludes strings containing 11 from the vertices. Any partial cube G embeds into some \( \varGamma _d \) , defining Fibonacci dimension, \( \textrm{fdim}(G) \) , as the minimum of such d. It holds \( \textrm{idim}(G) \le \textrm{fdim}(G) \le 2 \cdot \textrm{idim}(G) - 1 \) . While \( \textrm{idim}(G) \) is computable in polynomial time, check whether \( \textrm{idim}(G) = \textrm{fdim}(G) \) is NP-complete. We survey the properties of partial cubes and Generalized Fibonacci Cubes and present a new family of graphs G for which \( \textrm{idim}(G) = \textrm{fdim}(G) \) . We conclude with some open problems.

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Partial Cubes and Fibonacci Dimension: Insights and Perspectives

  • 会议论文

  • 发表日期：

    2026年

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  • Marcella Anselmo,
  • Dora Giammarresi,
  • Maria Madonia,
  • Sabrina Mantaci

摘要

A partial cube is a graph G that can be isometrically embedded into a hypercube \( Q_k \) , with the minimum of such k called the isometric dimension, \(\textrm{idim}(G)\) , of G. A Fibonacci cube \( \varGamma _k \) excludes strings containing 11 from the vertices. Any partial cube G embeds into some \( \varGamma _d \) , defining Fibonacci dimension, \( \textrm{fdim}(G) \) , as the minimum of such d. It holds \( \textrm{idim}(G) \le \textrm{fdim}(G) \le 2 \cdot \textrm{idim}(G) - 1 \) . While \( \textrm{idim}(G) \) is computable in polynomial time, check whether \( \textrm{idim}(G) = \textrm{fdim}(G) \) is NP-complete. We survey the properties of partial cubes and Generalized Fibonacci Cubes and present a new family of graphs G for which \( \textrm{idim}(G) = \textrm{fdim}(G) \) . We conclude with some open problems.