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

• Partition resolving set
• edge partition dimension
• partition dimension
• mixed partition dimension
• hexagonal network