The Mixed Partition Dimension: A New Resolvability Parameter in Graph Theory
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 strategie...
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| Main Authors: | , , , , |
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| Format: | Article |
| Language: | English |
| Published: |
IEEE
2025-01-01
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| Series: | IEEE Access |
| Subjects: | |
| Online Access: | https://ieeexplore.ieee.org/document/10856713/ |
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| Summary: | 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 <inline-formula> <tex-math notation="LaTeX">$R = \{W_{1}, W_{2}, {\dots }, W_{k}\}$ </tex-math></inline-formula> be a partition of the vertex set <inline-formula> <tex-math notation="LaTeX">$V(G)$ </tex-math></inline-formula> of a graph <inline-formula> <tex-math notation="LaTeX">$G = (V, E)$ </tex-math></inline-formula>, where <inline-formula> <tex-math notation="LaTeX">$W_{1} \cup W_{2} \cup {\dots } \cup W_{k} = V(G)$ </tex-math></inline-formula> and <inline-formula> <tex-math notation="LaTeX">$ W_{i} \cap W_{j} = \emptyset ~ \text {for}~ i \neq j$ </tex-math></inline-formula>. Each subset <inline-formula> <tex-math notation="LaTeX">$W_{i}$ </tex-math></inline-formula> is non-empty, mutually disjoint, and collectively covers all vertices. The partition set <inline-formula> <tex-math notation="LaTeX">${R}_{mp}$ </tex-math></inline-formula> is called mixed resolving partition set if it satisfies the condition. For any two distinct vertices <inline-formula> <tex-math notation="LaTeX">$u, v \in V(G)$ </tex-math></inline-formula>, there exists <inline-formula> <tex-math notation="LaTeX">$W_{i} \in R$ </tex-math></inline-formula> such that: <inline-formula> <tex-math notation="LaTeX">$d(u, W_{i}) \neq d(v, W_{i})$ </tex-math></inline-formula>, for any two distinct edges <inline-formula> <tex-math notation="LaTeX">$e_{1}, e_{2} \in E(G)$ </tex-math></inline-formula>, there exists <inline-formula> <tex-math notation="LaTeX">$W_{i} \in R$ </tex-math></inline-formula> such that: <inline-formula> <tex-math notation="LaTeX">$d(e_{1}, W_{i}) \neq d(e_{2}, W_{i})$ </tex-math></inline-formula> and for any vertex <inline-formula> <tex-math notation="LaTeX">$u \in V(G)$ </tex-math></inline-formula> and edge <inline-formula> <tex-math notation="LaTeX">$e \in E(G)$ </tex-math></inline-formula>, there exists <inline-formula> <tex-math notation="LaTeX">$W_{i} \in R$ </tex-math></inline-formula> such that: <inline-formula> <tex-math notation="LaTeX">$d(u, {W}_{i}) \neq d(e, W_{i})$ </tex-math></inline-formula>. The mixed partition dimension of G is the minimum number of subsets in a mixed resolving partition set <inline-formula> <tex-math notation="LaTeX">${R}_{mp}$ </tex-math></inline-formula>. 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. |
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| ISSN: | 2169-3536 |