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arXiv · 1110.5289

On the partition dimension of trees

Juan A. Rodriguez-Velazquez·Ismael G. Yero·Magdalena Lemanska
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Abstract

Given an ordered partition $Π=\{P_1,P_2, ...,P_t\}$ 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 $Π$ is the vector $r(v|Π)=(d(v,P_1),d(v,P_2),...,d(v,P_t))$, where $d(v,P_i)$ represents the distance between the vertex $v$ and the set $P_i$. A partition $Π$ 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 $u,v\in V$, $r(u|Π)\ne r(v|Π)$. 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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Juan A. Rodriguez-Velazquez, Ismael G. Yero, Magdalena Lemanska. 2011-10-24. On the partition dimension of trees. https://doi.org/10.1016/j.dam.2013.09.026

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