# Bled'11 - 7th Slovenian International Conference on Graph Theory

19-25 June 2011
Bled, Slovenia
Europe/Ljubljana timezone
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# On the partition dimension of corona graphs

Presented by Mr. Ismael GONZALEZ YERO
Type: Oral presentation
Track: Metric Graph Theory

## Content

Given a set of vertices $S=\{v_1,v_2,...,v_k\}$ of a connected graph $G$, the metric representation of a vertex $v$ of $G$ with respect to $S$ is the vector $r(v|S)=(d(v,v_1),d(v,v_2),...,d(v,v_k))$, where $d(v,v_i)$, $i\in \{1,...,k\}$ denotes the distance between $v$ and $v_i$. $S$ is a resolving set of $G$ if for every pair of vertices $u,v$ of $G$, $r(u|S)\ne r(v|S)$. The metric dimension $dim(G)$ of $G$ is the minimum cardinality of any resolving set of $G$. Given an ordered partition $\Pi =\{P_1,P_2, ...,P_t\}$ of vertices of a connected graph $G$, the partition representation of a vertex $v$ of $G$, with respect to the partition $\Pi$ is the vector $r(v|\Pi)=(d(v,P_1),d(v,P_2),...,d(v,P_t))$, where $d(v,P_i)$, $1\leq i\leq t$, represents the distance between the vertex $v$ and the set $P_i$, that is $d(v,P_i)=\min_{u\in P_i}\{d(v,u)\}$. $\Pi$ is a resolving partition for $G$ if for every pair of vertices $u,v$ of $G$, $r(u|\Pi)\ne r(v|\Pi)$. The partition dimension $pd(G)$ of $G$ is the minimum number of sets in any resolving partition for $G$. In this work we study the partition dimension of corona graphs. Particularly, we obtain some bounds on the partition dimension, we give some relationships between the metric dimension and partition dimension and we deduce some exact values of some particular cases of corona graphs.

## Place

Location: Bled, Slovenia
Address: Best Western Hotel Kompas Bled

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