On the OBDD representation of some graph classes

Abstract

Ordered Binary Decision Diagrams (OBDDs) are a popular data structure for Boolean functions. Given the rapid growth of application-based networks, an heuristic approach to deal with very large structured graphs are implicit OBDD-based graph algorithms. Vertices of an input graph are binary encoded and the edge set is represented by its characteristic function. Since OBDDs are able to take advantage of the presence of regular substructures, this approach leads sometimes to sublinear graph representations. By simple counting arguments it is easy to see that almost all graphs on N vertices cannot be represented by OBDDs of polylogarithmic size with respect to N. On the other hand, very simply structured graphs like grid graphs have small OBDD representations. Here, the investigation for which significant graph classes succinct OBDD representations are possible is continued. Using a new method how to use the structure of the adjacency matrix representation of a graph to count subfunctions of the characteristic Boolean function of its edge set, known upper bounds on the OBDD size of interval graphs are improved. Furthermore, the OBDD size for interval bigraphs, (bi)convex graphs, bipartite permutation graphs, chain graphs, threshold graphs, and trees is analyzed. Except for interval graphs, interval bigraphs, and convex graphs the presented bounds are tight. Finally, regular graphs are considered and examples are presented with small and large OBDD size. Moreover, a variable ordering often used in the implicit setting is investigated.