A dynamic implicit adjacency labelling scheme for line graphs

Abstract

As defined by Muller (Muller, Ph.D. thesis, Georgia Tech, 1988) and Kannan, Naor, and Rudich (Kannan et al., SIAM J Disc Math, 1992), an adjacency labelling scheme labels the vertices of a graph so the adjacency of two vertices can be deduced implicitly from their labels. In general, the labels used in adjacency labelling schemes cannot be tweaked to reflect small changes in the graph. Motivated by the necessity for further exploration of dynamic (implicit) adjacency labelling schemes we introduce the concept of error detection, discuss metrics for judging the quality of such dynamic schemes, and develop a dynamic scheme for line graphs that allows the addition and deletion of vertices and edges. The labels used in this scheme require O(log n) bits and updates can be performed in O(e) time, where e is the number of edges added to or deleted from the line graph. This compares to the best known (static) adjacency labelling scheme for line graphs which uses O(log n) bit labels and requires Θ(n) time to generate a labelling even when provided with the line graph representation. © Springer-Verlag Berlin Heidelberg 2005.