Ordered Binary Decision Diagrams (OBDDs for short) are popular dynamic data structures for Boolean functions. In some modern applications, we have to handle such huge graphs that the usual explicit representations by adjacency lists or adjacency matrices are infeasible. To deal with such huge graphs, OBDD-based graph representations and algorithms have been investigated. Although the size of OBDD representations may be large in general, it is known to be small for some special classes of graphs. In this paper, we show upper bounds and lower bounds of the size of OBDDs representing some intersection graphs such as bipartite permutation graphs, biconvex graphs, convex graphs, (2-directional) orthogonal ray graphs, and permutation graphs.

Ordered binary decision diagrams (OBDDs) have been widely used in many CAD applications as efficient data structures for representing and manipulating Boolean functions. For the efficient use of the OBDD, it is essential to find a good variable order, because the size of the OBDD heavily depends on its variable order. Dynamic variable reordering is a promising solution to the variable ordering problem of the OBDD. Dynamic variable reordering with the sifting algorithm is especially effective in minimizing the size of the OBDD and reduces the need to find a good initial variable order. However, it is very time-consuming for practical use. In this paper, we propose two new implementation techniques for fast dynamic variable reordering. One of the proposed techniques reduces the number of variable swaps by using the lower bound of the OBDD size, and the other accelerates the variable swap itself by recording the node states before the swap and the pivot nodes of the swap. By using these new techniques, we have achieved the speed-up ranging from 2.5 to 9.8 for benchmark circuits. These techniques have reduced the disadvantage of dynamic variable reordering and have made it more attractive for users.