Bounds of Fractional Domination in Space-Filling Curves

초록

Among space-filling geometric constructions, Peano and Hilbert graphs exhibit unique properties as recursively generated structures with rapidly increasing order and high levels of self-similarity. Although domination parameters in graph theory have been extensively studied, the fractional domination number of recursively generated space-filling graphs remains unexplored. This gap is particularly significant because the recursive growth and self-similarity of these graphs require efficient strategies for coverage and control. Fractional domination provides a flexible framework for optimizing resource allocation by allowing partial vertex contributions, making it well-suited for such complex structures. In this work, we present a precise graph-theoretic formulation for the n iterations of Peano and Hilbert graphs and investigate their fractional domination number. The fractional domination number is defined as the minimum total weight assigned to vertices, where each weight lies in the interval [0,1] , such that for every vertex, the sum of weights over its closed neighborhood is at least one. We derive closed-form expressions for these parameters and analyze their behavior under vertex addition, vertex deletion, and edge deletion. Additionally, we extend the study to Cartesian products of space-filling graphs with related graph classes. The theoretical results are supported by computational implementations in Java and Python, enabling efficient evaluation for large-scale iterations. These findings enhance the understanding of domination in self-similar networks and provide insights for optimization in distributed and networked systems.

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