Graph Clustering I." Cycles of Cliques (Extended Abstract)* F. J. Brandenburg Lehrstuht fiir Informatik Universit/it Passau 94030 Passan, Germany emaih brandenb@informatik.uni-passau.de Abstract. A graph is a cycle of cliques, if its set of vertices can be partitioned into clusters, such that each cluster is a clique and the cliques form a cycle. Then there is a partition of the set of edges into inner edges of the cliques and interconnection edges between the clusters. Cycles of cliques are a special instance of two-level clustered graphs. Such graphs are drawn by a two phase method: draw the top level graph and then browse into the clusters. In general, it is NP-hard whether or not a graph is a two-level clustered graph of a particular type, e.g. a clique or a planar graph or a triangle of cliques. However, it is efficiently solvable whether or not a graph is a path of cliques or is a large cycle of cliques. Introduction Graph clustering is a new direction in graph drawing. Several winners of last year's graph drawing competition have used this technique [9]. The Circular Library in the Graph Layout Toolkit of Tom Sawyer Software [5] clusters graphs and then displays them e.g. as a circle of circles. There is a general need for clustering techniques. As the amount of informa- tion to be visualized becomes larger, more structure is needed on top of the classical graph model. There is the need for abstraction and reduction. Flat graphs no longer suffice. When graphs become huge, classical graph drawing algorithms behave poorly or even fail. The further dit~cutty comes from the in- herent complexity of large graphs. The computational complexity of the graph drawing algorithms is directly effected by the size of the graphs. There is a need for etficient algorithms in graph drawing. Here, graph clustering brings us a step forward. Now, time consuming graph drawing algorithms can be applied to small portions only, and the overall running time still remains satisfactory. However, this can work only, if the partition into clusters can be computed efficiently. * Partially supported by the Deutsche Forschungsgemeinschaft, Forschungsschwer- punkt "Effiziente Algorithmen ffir diskrete Probleme und ihre Anwendungen".