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{\bf Ross J. Kang, L\'aszl\'o Lov\'asz, Tobias M\"uller and Edward R. Scheinerman}
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{\bf Dot Product Representations of Planar Graphs}
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A graph $G$ is a \emph{$k$-dot product graph} if there exists a vector labelling $u: V(G) \to \mathbb{R}^k$ such that $u(i)^{T}u(j) \geq 1$ if and only if $ij \in E(G)$. Fiduccia, Scheinerman, Trenk and Zito~[{\em Discrete Math.}, 1998] asked whether
every planar graph is a $3$-dot product graph. We show that the answer is ``no''.
On the other hand, every planar graph is a $4$-dot product graph. We also answer the corresponding questions for planar graphs of prescribed girth and for outerplanar graphs.

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