This paper presents the problem of drawing a graph in the plane so that edges appear as straight lines and the minimum angle formed by any pair of incident edges is maximized. The resolution of a layout is defined to be the size of the minimum angle formed by incident edges of the graph, and the resolution of a graph to be the maximum resolution of any layout of the graph. The resolution R of a graph is characterized in terms of the maximum node degree d of the graph by proving that $\Omega (\frac{1}{{d^2 }}) \leqslant R \leqslant \frac{{2\pi }}{d}$ for any graph. Moreover, it is proved that $R = \Theta (\frac{1}{d})$ for many graphs including planar graphs, complete graphs, hypercubes, multidimensional meshes and tori, and other special networks. It is also shown that the problem of deciding if $R = \frac{{2\pi }}{d}$ for a graph is NP-hard for $d = 4$, and by using a counting argument that $R = O(\frac{{\log d}}{{d^2 }})$ for many graphs.

The following packing problem arises in connection with lettering of maps: Given n distinct points pl, p2,.... pn in the plane, determine the supremum uoPi of all reals U, such that there are n pan-wise dtsjomt, axis-parallel, closed squares Ql, Q2,.... Qn of side-length u, where each pi ts a corner of Qi. Note that — by using afine transformation — the problem is equivalent to the case when we want largest homothetic cop~es of a jized rectangle or parallelogram tnstead of equal ly-szzed squares. In the cartographic application, the points are items (groundwater-drillho les etc.) and the squares are places for labels associated with these items (sulphate concentration etc.). An algorithm is presented, that in O(n log n] time either produces a solution, that is guaranteed to be at least half as large as the supremum. This is optimal, m the sense that the corresponding decision problem is NP-complete, no po[ynomzal approximation algorithm with a guaranteed factor ezceedmg ~ exwts, provided that P # AfP; and there M also a lower bound of C2(n log n) for the running time. 1

The problem of drawing a graph in the plane so that edges appear as straight lines and the minimum angle formed by any pair of incident edges is maximized is studied. The resolution of a layout is defined to be the size of the minimum angle formed by incident edges of the graph, and the resolution of a graph is defined to be the maximum resolution of any layout of the graph. The resolution R of a graph is characterized in terms of the maximum node degree d of the graph by proving that Omega (1/d/sup 2/)>