Dot Product Representations of Planar Graphs
نویسندگان
چکیده
A graph G is a k-dot product graph if there exists a vector labelling u : V (G)→ Rk such that u(i)Tu(j) ≥ 1 if and only if ij ∈ E(G). Fiduccia, Scheinerman, Trenk and Zito [4] 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.
منابع مشابه
Dot Product Representations of Graphs
We introduce the concept of dot product representations of graphs, giving some motivations as well as surveying the previously known results. We extend these representations to more general fields, looking at the complex numbers, rational numbers, and finite fields. Finally, we study the behavior of dot product representations in field extensions.
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