Article

# On the partition dimension of trees

10/2011;
Source: arXiv

ABSTRACT Given an ordered partition $\Pi =\{P_1,P_2, ...,P_t\}$ of the vertex set $V$
of a connected graph $G=(V,E)$, the \emph{partition representation} of a vertex
$v\in V$ 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 $d(v,P_i)$ represents the
distance between the vertex $v$ and the set $P_i$. A partition $\Pi$ of $V$ is
a \emph{resolving partition} of $G$ if different vertices of $G$ have different
partition representations, i.e., for every pair of vertices $u,v\in V$,
$r(u|\Pi)\ne r(v|\Pi)$. The \emph{partition dimension} of $G$ is the minimum
number of sets in any resolving partition of $G$. In this paper we obtain
several tight bounds on the partition dimension of trees.

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12 Nov 2012

### Keywords

\emph{partition dimension}

\emph{partition representation}

\emph{resolving partition}

bounds

different vertices

partition $\Pi$

sets

vertex

vertex $v$