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IEEE Access
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The Mixed Partition Dimension: A New Resolvability Parameter in Graph Theory

Authors: Siti Norziahidayu Amzee Zamri; Sikander Ali; Muhammad Azeem; Husam A. Neamah; Bandar Almohsen;

The Mixed Partition Dimension: A New Resolvability Parameter in Graph Theory

Abstract

In this article, we introduce a novel graph-theoretical parameter called the mixed partition dimension and apply it to the path graph and the hexagonal network. This parameter builds on the concept of resolvability in graphs, integrating vertex-based partition dimensions with edge-oriented strategies to characterize the complexity of graph structures. It is the extension of the mixed metric dimension and partition dimension. Suppose Let $R = \{W_{1}, W_{2}, {\dots }, W_{k}\}$ be a partition of the vertex set $V(G)$ of a graph $G = (V, E)$ , where $W_{1} \cup W_{2} \cup {\dots } \cup W_{k} = V(G)$ and $ W_{i} \cap W_{j} = \emptyset ~ \text {for}~ i \neq j$ . Each subset $W_{i}$ is non-empty, mutually disjoint, and collectively covers all vertices. The partition set ${R}_{mp}$ is called mixed resolving partition set if it satisfies the condition. For any two distinct vertices $u, v \in V(G)$ , there exists $W_{i} \in R$ such that: $d(u, W_{i}) \neq d(v, W_{i})$ , for any two distinct edges $e_{1}, e_{2} \in E(G)$ , there exists $W_{i} \in R$ such that: $d(e_{1}, W_{i}) \neq d(e_{2}, W_{i})$ and for any vertex $u \in V(G)$ and edge $e \in E(G)$ , there exists $W_{i} \in R$ such that: $d(u, {W}_{i}) \neq d(e, W_{i})$ . The mixed partition dimension of G is the minimum number of subsets in a mixed resolving partition set ${R}_{mp}$ . This parameter provides a unified measure of a graph’s complexity by accounting for both vertex and edge distinguishability, offering new insights into the structure of complex networks.

Keywords

mixed partition dimension, Partition resolving set, edge partition dimension, partition dimension, Electrical engineering. Electronics. Nuclear engineering, hexagonal network, TK1-9971

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citations
This is an alternative to the "Influence" indicator, which also reflects the overall/total impact of an article in the research community at large, based on the underlying citation network (diachronically).
BIP!Citations provided by BIP!
popularity
This indicator reflects the "current" impact/attention (the "hype") of an article in the research community at large, based on the underlying citation network.
BIP!Popularity provided by BIP!
influence
This indicator reflects the overall/total impact of an article in the research community at large, based on the underlying citation network (diachronically).
BIP!Influence provided by BIP!
impulse
This indicator reflects the initial momentum of an article directly after its publication, based on the underlying citation network.
BIP!Impulse provided by BIP!
0
Average
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Average
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