On the OBDD Size for Graphs of Bounded Tree- and Clique-Width
Klaus Meer and Dieter Rautenbach
In: The 2nd International Workshop on Parameterized and Exact Computation, 13-15 Sep 2006, Zuerich, Switzerland.

## Abstract

We study the size of OBDDs (ordered binary decision diagrams) for representing the adjacency function $f_G$ of a graph $G$ on $n$ vertices. Our results are as follows: - for graphs of bounded tree-width there is an OBDD of size $O(\log{n})$ for $f_G$ that uses encodings of size $O(\log{n})$ for the vertices; - for graphs of bounded clique-width there is an OBDD of size $O(n)$ for $f_G$ that uses encodings of size $O(n)$ for the vertices; - for graphs of bounded clique-width such that there is a \emph{reduced term} for $G$ (to be defined below) that is balanced with depth $O(\log{n})$ there is an OBDD of size $O(n)$ for $f_G$ that uses encodings of size $O(\log{n})$ for the vertices; - for cographs, i.e. graphs of clique-width at most 2, there is an OBDD of size $O(n)$ for $f_G$ that uses encodings of size $O(\log{n})$ for the vertices. This last result improves a recent result by Nunkesser and Woelfel \cite{Nunkesser}.