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