- Author:
- David Morgan
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 a graph such that 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.
First studied by Brodal and Fagerberg [Brodal and Fagerberg, LNCS 1663, 1999], a dynamic adjacency labelling scheme is an adjacency labelling scheme that requires only small adjustments to the vertex labels when a small change is made to the graph. Motivated by the necessity for further exploration of dynamic adjacency labelling schemes, we introduce the concept of error-detection, discuss metrics for judging the quality of dynamic schemes, and develop error-detecting fully dynamic schemes for several classes of graphs.
Our dynamic scheme for line graphs uses O(log n ) bit labels and updates in O( e ) time, where e is the number of edges added to, or deleted from, the line graph. As well, our dynamic scheme for proper interval graphs uses O(log n ) bit labels and handles all operations in O( n ) time.
We also develop a O( r log n ) bit/label dynamic adjacency labelling scheme for r-minoes , which are graphs with no vertex in more than r maximal cliques. Edge addition and deletion can be handled in O( r 2 D ) time, vertex addition in O( r 2 e 2 ) time, and vertex deletion in O( r 2 e ) time, where D is the maximum degree of the vertices in the original graph and e is the number of edges added to, or deleted from, the original graph.
Similar to this dynamic scheme for r -minoes, we develop a O( r log n ) bit/label dynamic adjacency labelling scheme for r-bics , which are graphs with no vertex in more than r maximal bicliques. Edge addition and deletion, as well as vertex deletion, can be handled in O( r 2 B ) time, and vertex addition in O( r 2 n B ) time, where B is the size of the largest biclique in the original graph.
Our dynamic labelling schemes for r -minoes and r -bics lead to O( r 2 n 3 ) time recognition algorithms for both of these classes.
Index Terms
- Dynamic adjacency labelling schemes
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