# On the Strong Partition Dimension of Graphs

• Ismael González Yero
Keywords: Strong resolving set, strong metric dimension, strong resolving partition, strong partition dimension, strong resolving graph

### Abstract

We present a new style of metric generator in graphs. Specifically we introduce a metric generator based on a partition of the vertex set of a graph. The sets of the partition will work as the elements which will uniquely determine the position of each single vertex of the graph. A set $W$ of vertices of a connected graph $G$ strongly resolves two different vertices $x,y\notin W$ if either $d_G(x,W)=d_G(x,y)+d_G(y,W)$ or $d_G(y,W)=d_G(y,x)+d_G(x,W)$, where $d_G(x,W)=\min\left\{d(x,w)\;:\;w\in W\right\}$. An ordered vertex partition $\Pi=\left\{U_1,U_2,...,U_k\right\}$ of a graph $G$ is a strong resolving partition for $G$ if every two different vertices of $G$ belonging to the same set of the partition are strongly resolved by some set of $\Pi$. A strong resolving partition of minimum cardinality is called a strong partition basis and its cardinality the strong partition dimension. In this article we introduce the concepts of strong resolving partition and strong partition dimension and we begin with the study of its mathematical properties.

Published
2014-07-25
Article Number
P3.14