Drawing Graphs in the Plane with High Resolution

In thb paper, we study the problem of drawing a graph in the plane so that edger appear aa straight lines and so that the "um angle formed by any pair of incident edges is " k e d . We define the rerolution of a layout to be the s t e of the minimum &e fonned by incident edges of the graph, and the resolution of a graph to be the d u m raohtion of any layout of the graph. We characterise the resolution R of a graph in tumr of the maximom node degree d of the graph by proving that n($) < R < 9 for any graph. Moreover, we prove that R = e( f ) for many graphs including planar graphs, complete graphs, hypercubes, multidimensional meshes and tori, and other special networka. We also show that the problem of deciding if R = ?Lf for a graph to show that R = O( v) for many graphs. is NP-hard for d = 4, and wc usc a cowting argument P u t i d @ supported by the ESPRIT XI Buic Ftmcuch Actions Program d the EC under contrwt No. 80% (project ALCOM). Supported by the CLEAR Ccnter at UTD, UTD proposal #8?0040. Supported hy the DPG, SonduTorwhungsbucich 124, TeiL projekt E2 (-I-Entwurhmethoden und Parallelit&). Supported by Air P o m Contract APSOR8%0721, DARPA Contract N00014-87-K826, Army Contrut DMGOS-8S Supported by the Ponds sur Plrderung der WiucnschdtlC chen Forrehung (PWP), Project ss2/ol. K-Oln, md the C~CW Ccnta at UTD. 'Institut fur MathcmatiL Technics University of Grar Kopemikusgauc 24 A-8010 Graa Austria