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Local Multiset Dimension of Amalgamation Graphs.

Ridho Alfarisi1,2, Liliek Susilowati1, Dafik Dafik3

  • 1Mathematics, Universitas Airlangga, Surabaya, Surabaya, 68121, Indonesia.

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|June 5, 2024
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Summary

This study introduces the concept of local multiset dimension in graphs and establishes an upper bound for amalgamated graphs. Wheel graphs are identified as examples that achieve this upper bound for local resolving sets.

Keywords:
local m-resolving set; local multiset dimension; amalgamation graph.

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Area of Science:

  • Graph Theory
  • Discrete Mathematics
  • Combinatorics

Background:

  • Introduces the resolving set problem in graph theory, focusing on vertex representations based on shortest path distances.
  • Defines a local m-resolving set and the local multiset dimension (md(G)) as the minimum cardinality of such a set.

Purpose of the Study:

  • To determine the upper bound of the local multiset dimension for amalgamated graphs.
  • To calculate the exact local multiset dimension for specific families of graphs, including paths, cycles, and wheel graphs.

Main Methods:

  • Employs a pure research design with an exploratory approach.
  • Involves selecting special graphs formed by amalgamation, defining vertex and edge sets, and determining vertex representations.
  • Utilizes theorem proving to establish bounds and exact values for the local multiset dimension.

Main Results:

  • Establishes an upper bound for the local multiset dimension of amalgamated graphs: md(Amal(G, v, m)) ≤ m * md(G).
  • Determines the exact local multiset dimension for specific graph families, such as paths (md(Amal(P, v, m)) = 1) and wheel graphs.

Conclusions:

  • Successfully derived an upper bound for the local multiset dimension of amalgamated graphs.
  • Identified wheel graphs as examples that attain this upper bound, demonstrating specific cases where the maximum is reached.