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DISTINCT DISTANCES IN GRAPH DRAWINGS
, 2008
"... The distancenumber of a graph G is the minimum number of distinct edgelengths over all straightline drawings of G in the plane. This definition generalises many wellknown concepts in combinatorial geometry. We consider the distancenumber of trees, graphs with no K − 4minor, complete bipartite g ..."
Abstract

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The distancenumber of a graph G is the minimum number of distinct edgelengths over all straightline drawings of G in the plane. This definition generalises many wellknown concepts in combinatorial geometry. We consider the distancenumber of trees, graphs with no K − 4minor, complete bipartite graphs, complete graphs, and cartesian products. Our main results concern the distancenumber of graphs with bounded degree. We prove that nvertex graphs with bounded maximum degree and bounded treewidth have distancenumber in 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 distancenumber. Similarly, we prove that there exist graphs with maximum degree 5 and arbitrarily large distancenumber. Moreover, as ∆ increases the existential lower bound on the distancenumber of ∆regular graphs tends to Ω(n0.864138). 1