The Journal of Supercomputing, 31, 265–279, 2005 C  2005 Springer Science + Business Media, Inc. Manufactured in The Netherlands. Single-Row Transformation of Complete Graphs SHAHARUDDIN SALLEH Department of Mathematics, Faculty of Science, Universiti Teknologi Malaysia, 81310 Johor Bahru, Malaysia STEPHAN OLARIU Department of Computer Science, Old Dominion University, Norfolk, VA 23529, USA BAHROM SANUGI MOHD ISMAIL ABD AZIZ Department of Mathematics, Faculty of Science, Universiti Teknologi Malaysia, 81310 Johor Bahru, Malaysia Abstract. A complete graph is a fully-connected graph where every node is adjacent to all other nodes in the graph. Very often, many applications in science and engineering are reducible to this type of graph. Hence, a simplified form of a complete graph contributes in providing the solutions to these problems. In this paper, we present a technique for transforming a complete graph into a single-row routing problem. Single-row routing is a classical technique in the VLSI design that is known to be NP-complete. We solved this problem earlier using a method called ESSR, and, the same technique is applied to the present work to transform a complete graph into its single-row routing representation. A parallel computing model is proposed which contributes in making the problem modular and scalable. We also discuss the application of this work on the channel assignment problem in the wireless cellular telephone networks. Keywords: complete graph, single-row routing, simulated annealing and channel assignments 1. Introduction In many cases, problems in engineering and other technical problems can be represented as problems in graph theory. A problem of this nature is said to be reducible to the form of vertices and links of a graph, and the solution to the problem can be obtained by solving the graph problem. Furthermore, several solutions to the problems in graph theory have found their roots in some well-known prototype problems, such as the traveling salesman problem, the shortest path problem and the minimum spanning tree problem. Solutions to these problems are provided in the form of dynamic programming techniques, mathematical programming and heuristics. Most of these prototype problems have been proven to be NP- complete and, therefore, no absolute solutions to the problems are established. However, their reduction to the form of graphs have, in some ways, simplified their complexity and pave way for further improvement to their solutions. In this paper, we study the relationship between a complete graph and its single-row representation. A complete graph is a graph where every vertex in the graph is adjacent to all other vertices. Single-row routing is a classical problem about finding an optimum routing from a set of terminals, or nodes, arranged in a single-row in the printed circuit boards (PCB) design. In the Very Large Scale Integration (VLSI) technology, the terminals