Constant memory routing in quasi-planar and quasi-polyhedral graphs

We address the problem of online route discovery for it class of graphs that call be embedded either in two or in three-dimensional space. In two dimensions we propose the class of quasi-planar graphs and in three dimensions the class of quasi-polyhedral graphs. In the former case such graphs are geometrically embedded in R-2 and have an underlying backbone that is planar with convex faces; however within each face arbitrary edges (with arbitrary crossings) are allowed. In the latter case, these graphs are geometrically embedded in R-3 and consist of a backbone of convex polyhedra and arbitrary edges within each polyhedron. In both cases we provide a routing algorithm that guarantees delivery. Our algorithms need only "remember" the source and destination nodes and one (respectively, two) reference nodes used to store information about the underlying face (respectively, polyhedron) currently being traversed. The existence of the backbone is used only in proof's of correctness of the routine algorithm; the particular choice is irrelevant and does not affect the behaviour of the algorithm. Crown Copyright (C) 2008 Published by Elsevier B.V. All rights reserved.