On the Optimal Layout of Planar Graphs with Fixed Boundary

Abstract

The optimal planar layout of planar graphs with respect to the $L_1 $- or $L_2 $-metric leads to NP-hard problems, if one assumes the nodes of the graph to be fixed in the plane (see [FiPa], [Be]).In this paper we consider the (optimal) layout of graphs with fixed boundary (i.e., graphs, where only the nodes of a given cycle of the graph have fixed positions in the plane). The investigated layouts are straight line embeddings in a continuous part of the plane; the cost of a layout is calculated with help of very general cost functions including the pth power of the usual Euclidean distance metric for $p = 2,3, \cdots $ (for short, $l_p $-metric).For a large class of graphs, which, for example, occur in chip layout problems as the abstract structure of switching circuits, we show the existence and uniqueness of the optimal layout.The main part of the paper is concerned with planar graphs. We get an interesting characterization of nonplanar layouts of planar graphs, which shows that the optimal layout of a ...