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 = {W1, W2, ·, Wk} be a partition of the vertex set V(G) of a graph G = (V, E) , where W1 ∪ W2 ∪ · ∪ Wk = V(G) and Wi ∩ Wj = ∅ for i ≢ j. Each subset Wi is non-empty, mutually disjoint, and collectively covers all vertices. The partition set Rmp is called mixed resolving partition set if it satisfies the condition. For any two distinct vertices u, v ϵ V(G) , there exists Wi ϵ R such that: d(u, Wi) ≢ d(v, Wi), for any two distinct edges e1, e2 ϵ E(G), there exists Wi ϵ R such that: d(e1, Wi) ≢ d(e2, Wi) and for any vertex u ϵ V(G) and edge e ϵ E(G), there exists Wi ϵ R such that: d(u, Wi) ≢ d(e, Wi). The mixed partition dimension of G is the minimum number of subsets in a mixed resolving partition set Rmp. 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.