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arXiv · 0804.3690

Distinct Distances in Graph Drawings

Paz Carmi·Vida Dujmović·Pat Morin·David R. Wood
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Abstract

The \emph{distance-number} of a graph $G$ is the minimum number of distinct edge-lengths over all straight-line drawings of $G$ in the plane. This definition generalises many well-known concepts in combinatorial geometry. We consider the distance-number of trees, graphs with no $K^-_4$-minor, complete bipartite graphs, complete graphs, and cartesian products. Our main results concern the distance-number of graphs with bounded degree. We prove that $n$-vertex graphs with bounded maximum degree and bounded treewidth have distance-number in $\mathcal{O}(\log n)$. To conclude such a logarithmic upper bound, both the degree and the treewidth need to be bounded. In particular, we construct graphs with treewidth 2 and polynomial distance-number. Similarly, we prove that there exist graphs with maximum degree 5 and arbitrarily large distance-number. Moreover, as $Δ$ increases the existential lower bound on the distance-number of $Δ$-regular graphs tends to $Ω(n^{0.864138})$.

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Paz Carmi, Vida Dujmović, Pat Morin, David R. Wood. 2008-04-23. Distinct Distances in Graph Drawings. https://arxiv.org/abs/0804.3690

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