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Certain Domination Parameters and its Resolving Version of Fractal Cubic Networks

S. Prabhu, A. K. Arulmozhi, M. Arulperumjothi

Abstract

Networks are designed to communicate, operate and allocate the tasks to the respective commodities. Operating the supercomputers became challenging, and it was handled by the network design commonly known as hypercube, denoted by $Q^n$. In a recent study, the hypercube networks were not enough to hold the parallel processors in the supercomputers. Thus, variants of hypercubes were discovered to produce an alternative to the hypercube. A new variant of the hypercube, the \textit{fractal cubic network}, can be used as the best alternative in the case of hypercubes, which was wrongly defined in [Eng. Sci. Technol. \textbf{18}(1) (2015) 32--41]. Arulperumjothi et al. recently corrected this definition and redefined the network in [Appl. Math. Comput. \textbf{452} (2023) 128037]. Our research investigates that the fractal cubic network is a \textit{rooted product} of two graphs. We try to determine its domination and resolving domination parameters, which could be applied to resource location and broadcasting-related problems.

Certain Domination Parameters and its Resolving Version of Fractal Cubic Networks

Abstract

Networks are designed to communicate, operate and allocate the tasks to the respective commodities. Operating the supercomputers became challenging, and it was handled by the network design commonly known as hypercube, denoted by . In a recent study, the hypercube networks were not enough to hold the parallel processors in the supercomputers. Thus, variants of hypercubes were discovered to produce an alternative to the hypercube. A new variant of the hypercube, the \textit{fractal cubic network}, can be used as the best alternative in the case of hypercubes, which was wrongly defined in [Eng. Sci. Technol. \textbf{18}(1) (2015) 32--41]. Arulperumjothi et al. recently corrected this definition and redefined the network in [Appl. Math. Comput. \textbf{452} (2023) 128037]. Our research investigates that the fractal cubic network is a \textit{rooted product} of two graphs. We try to determine its domination and resolving domination parameters, which could be applied to resource location and broadcasting-related problems.

Paper Structure

This paper contains 5 sections, 32 theorems, 4 equations, 12 figures, 1 table.

Table of Contents

  1. Introduction and Motivation
  2. Resolving and Dominating Sets
  3. Fractal Cube: A Fascinating Variant of Hypercube
  4. Main Results
  5. Conclusion

Key Result

Theorem 1

PrFlAr18 Let $\Gamma$ be a connected graph with twin sets $T_k$, $1 \leq k \leq p$, then $\dim(\Gamma) \geq \sum_{i=1}^{p}|T_k|-p$.

Figures (12)

  • Figure 1: Resolving set $R$
  • Figure 2: Various dimensions of hypercubes
  • Figure 3: Various dimensions of $FCN$
  • Figure 4: (a)$\Gamma$; (b)TD and CD set of $\Gamma$; (c) Resolving set of $\Gamma$
  • Figure 5: DS and IDS of $FCN(2)$
  • ...and 7 more figures

Theorems & Definitions (53)

  • Theorem 1
  • Theorem 2
  • Theorem 3
  • Theorem 4
  • Theorem 5
  • Theorem 6
  • Theorem 7
  • Theorem 8
  • Theorem 9
  • Theorem 10
  • ...and 43 more