The Price of Order

We present tight bounds on the spanning ratio of a large family of ordered -graphs. A -graph partitions the plane around each vertex into disjoint cones, each having aperture . An ordered -graph is constructed by inserting the vertices one by one and connecting each vertex to the closest previously-inserted vertex in each cone. We show that for any integer , ordered -graphs with cones have a tight spanning ratio of . We also show that for any integer , ordered -graphs with cones have a tight spanning ratio of . We provide lower bounds for ordered -graphs with and cones. For ordered -graphs with and cones these lower bounds are strictly greater than the worst case spanning ratios of their unordered counterparts. These are the first results showing that ordered -graphs have worse spanning ratios than unordered -graphs. Finally, we show that, unlike their unordered counterparts, the ordered -graphs with 4, 5, and 6 cones are not spanners.