A degree sequence problem related to network design

Abstract

AbstractWe consider a combinatorial problem arising in the design and operation of lightwave networks. Nodes in such networks are equipped with tunable transmitters and receivers and communication occurs when the frequency of some transmitter is the same as that of a receiver. This technology enables us to update the network topology to respond to changes in traffic patterns. There are two main optimization problems related to this network structure, one being the design of a target graph more suitable to (future) traffic conditions, and the other being the problem of transforming the current network to this target network. This paper discusses the second problem, i.e., the transition phase when the modifications on the current graph are made through a sequence of intermediate connection networks. In particular, we move from one graph to another by swapping two independent edges in the current graph for two other independent edges not in the current graph, so that the union forms a four‐cycle. We characterize the sequence requiring the minimum number of intermediate graphs together with the necessary and sufficient conditions for the existence of such a sequence. We also develop upper and lower bounds on the length of a shortest sequence by formulating an integer program and solving its continuous relaxation to optimality. We also give an efficient algorithm for the case when the intermediate graphs are required to be connected. © 1994 by John Wiley & Sons, Inc.